Mathematical Problems in Engineering

2021

DOI: 10.1155/2021/9945114

|View full text |Cite

|

Sign up to set email alerts

|

Hadamard and Fejér–Hadamard Inequalities for $(α, h - m) - pWenyan Jia 1, Muhammad Yussouf 2, Ghulam Farid 3 et al.Abstract: In this paper, we introduce

α
,
h
−
m

−
p

-convex function and some related functions. By applying this generalized definition, new versions of Hada… Show moreHelp me understand this reportSearch citation statements .css-b62m3t-container{position:relative;box-sizing:border-box;} .css-7pg0cj-a11yText{z-index:9999;border:0;clip:rect(1px, 1px, 1px, 1px);height:1px;width:1px;position:absolute;overflow:hidden;padding:0;white-space:nowrap;} .css-1vom5ua-control{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;cursor:default;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-box-flex-wrap:wrap;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-box-pack:justify;-webkit-justify-content:space-between;justify-content:space-between;min-height:38px;outline:0!important;position:relative;-webkit-transition:all 100ms;transition:all 100ms;background-color:hsl(0, 0%, 100%);border-color:hsl(0, 0%, 80%);border-radius:12px;border-style:solid;border-width:1px;box-sizing:border-box;}.css-1vom5ua-control:hover{border-color:hsl(0, 0%, 70%);} .css-hlgwow{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;display:grid;-webkit-flex:1;-ms-flex:1;flex:1;-webkit-box-flex-wrap:wrap;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-overflow-scrolling:touch;position:relative;overflow:hidden;padding:2px 8px;box-sizing:border-box;} Order By: Relevance .css-19bb58m{visibility:visible;-webkit-flex:1 1 auto;-ms-flex:1 1 auto;flex:1 1 auto;display:inline-grid;grid-area:1/1/2/3;grid-template-columns:0 min-content;margin:2px;padding-bottom:2px;padding-top:2px;color:hsl(0, 0%, 20%);box-sizing:border-box;}.css-19bb58m:after{content:attr(data-value) " ";visibility:hidden;white-space:pre;grid-area:1/2;font:inherit;min-width:2px;border:0;margin:0;outline:0;padding:0;} .css-1wy0on6{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;-webkit-align-self:stretch;-ms-flex-item-align:stretch;align-self:stretch;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-shrink:0;-ms-flex-negative:0;flex-shrink:0;box-sizing:border-box;} .css-1xc3v61-indicatorContainer{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-transition:color 150ms;transition:color 150ms;color:hsl(0, 0%, 80%);padding:8px;box-sizing:border-box;}.css-1xc3v61-indicatorContainer:hover{color:hsl(0, 0%, 60%);} .css-8mmkcg{display:inline-block;fill:currentColor;line-height:1;stroke:currentColor;stroke-width:0;} Paper Sections .css-b62m3t-container{position:relative;box-sizing:border-box;} .css-7pg0cj-a11yText{z-index:9999;border:0;clip:rect(1px, 1px, 1px, 1px);height:1px;width:1px;position:absolute;overflow:hidden;padding:0;white-space:nowrap;} .css-13cymwt-control{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;cursor:default;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-box-flex-wrap:wrap;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-box-pack:justify;-webkit-justify-content:space-between;justify-content:space-between;min-height:38px;outline:0!important;position:relative;-webkit-transition:all 100ms;transition:all 100ms;background-color:hsl(0, 0%, 100%);border-color:hsl(0, 0%, 80%);border-radius:4px;border-style:solid;border-width:1px;box-sizing:border-box;}.css-13cymwt-control:hover{border-color:hsl(0, 0%, 70%);} .css-hlgwow{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;display:grid;-webkit-flex:1;-ms-flex:1;flex:1;-webkit-box-flex-wrap:wrap;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-overflow-scrolling:touch;position:relative;overflow:hidden;padding:2px 8px;box-sizing:border-box;} .css-1jqq78o-placeholder{grid-area:1/1/2/3;color:hsl(0, 0%, 50%);margin-left:2px;margin-right:2px;box-sizing:border-box;} Select... .css-19bb58m{visibility:visible;-webkit-flex:1 1 auto;-ms-flex:1 1 auto;flex:1 1 auto;display:inline-grid;grid-area:1/1/2/3;grid-template-columns:0 min-content;margin:2px;padding-bottom:2px;padding-top:2px;color:hsl(0, 0%, 20%);box-sizing:border-box;}.css-19bb58m:after{content:attr(data-value) " ";visibility:hidden;white-space:pre;grid-area:1/2;font:inherit;min-width:2px;border:0;margin:0;outline:0;padding:0;} .css-1wy0on6{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;-webkit-align-self:stretch;-ms-flex-item-align:stretch;align-self:stretch;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-shrink:0;-ms-flex-negative:0;flex-shrink:0;box-sizing:border-box;} .css-1u9des2-indicatorSeparator{-webkit-align-self:stretch;-ms-flex-item-align:stretch;align-self:stretch;width:1px;background-color:hsl(0, 0%, 80%);margin-bottom:8px;margin-top:8px;box-sizing:border-box;} .css-1xc3v61-indicatorContainer{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-transition:color 150ms;transition:color 150ms;color:hsl(0, 0%, 80%);padding:8px;box-sizing:border-box;}.css-1xc3v61-indicatorContainer:hover{color:hsl(0, 0%, 60%);} .css-8mmkcg{display:inline-block;fill:currentColor;line-height:1;stroke:currentColor;stroke-width:0;} Introduction 1 Citation Types Supporting 0 Mentioning 1 Contrasting 0 Year Published 2022202220242024 Publication Types .css-b62m3t-container{position:relative;box-sizing:border-box;} .css-7pg0cj-a11yText{z-index:9999;border:0;clip:rect(1px, 1px, 1px, 1px);height:1px;width:1px;position:absolute;overflow:hidden;padding:0;white-space:nowrap;} .css-13cymwt-control{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;cursor:default;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-box-flex-wrap:wrap;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-box-pack:justify;-webkit-justify-content:space-between;justify-content:space-between;min-height:38px;outline:0!important;position:relative;-webkit-transition:all 100ms;transition:all 100ms;background-color:hsl(0, 0%, 100%);border-color:hsl(0, 0%, 80%);border-radius:4px;border-style:solid;border-width:1px;box-sizing:border-box;}.css-13cymwt-control:hover{border-color:hsl(0, 0%, 70%);} .css-hlgwow{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;display:grid;-webkit-flex:1;-ms-flex:1;flex:1;-webkit-box-flex-wrap:wrap;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-overflow-scrolling:touch;position:relative;overflow:hidden;padding:2px 8px;box-sizing:border-box;} .css-1jqq78o-placeholder{grid-area:1/1/2/3;color:hsl(0, 0%, 50%);margin-left:2px;margin-right:2px;box-sizing:border-box;} Select... .css-19bb58m{visibility:visible;-webkit-flex:1 1 auto;-ms-flex:1 1 auto;flex:1 1 auto;display:inline-grid;grid-area:1/1/2/3;grid-template-columns:0 min-content;margin:2px;padding-bottom:2px;padding-top:2px;color:hsl(0, 0%, 20%);box-sizing:border-box;}.css-19bb58m:after{content:attr(data-value) " ";visibility:hidden;white-space:pre;grid-area:1/2;font:inherit;min-width:2px;border:0;margin:0;outline:0;padding:0;} .css-1wy0on6{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;-webkit-align-self:stretch;-ms-flex-item-align:stretch;align-self:stretch;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-shrink:0;-ms-flex-negative:0;flex-shrink:0;box-sizing:border-box;} .css-1u9des2-indicatorSeparator{-webkit-align-self:stretch;-ms-flex-item-align:stretch;align-self:stretch;width:1px;background-color:hsl(0, 0%, 80%);margin-bottom:8px;margin-top:8px;box-sizing:border-box;} .css-1xc3v61-indicatorContainer{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-transition:color 150ms;transition:color 150ms;color:hsl(0, 0%, 80%);padding:8px;box-sizing:border-box;}.css-1xc3v61-indicatorContainer:hover{color:hsl(0, 0%, 60%);} .css-8mmkcg{display:inline-block;fill:currentColor;line-height:1;stroke:currentColor;stroke-width:0;} Article 5 Relationship Self Cite 1 Independent 4 Authors Journals OK Export report Add to dashboard Cited by 13 publication s (1 citation statement) References 19 publication s Supporting 0 Mentioning 1 Contrasting 0 .css-b62m3t-container{position:relative;box-sizing:border-box;} .css-7pg0cj-a11yText{z-index:9999;border:0;clip:rect(1px, 1px, 1px, 1px);height:1px;width:1px;position:absolute;overflow:hidden;padding:0;white-space:nowrap;} .css-1vom5ua-control{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;cursor:default;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-box-flex-wrap:wrap;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-box-pack:justify;-webkit-justify-content:space-between;justify-content:space-between;min-height:38px;outline:0!important;position:relative;-webkit-transition:all 100ms;transition:all 100ms;background-color:hsl(0, 0%, 100%);border-color:hsl(0, 0%, 80%);border-radius:12px;border-style:solid;border-width:1px;box-sizing:border-box;}.css-1vom5ua-control:hover{border-color:hsl(0, 0%, 70%);} .css-hlgwow{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;display:grid;-webkit-flex:1;-ms-flex:1;flex:1;-webkit-box-flex-wrap:wrap;-webkit-flex-wrap:wrap;-ms-flex-wrap:wrap;flex-wrap:wrap;-webkit-overflow-scrolling:touch;position:relative;overflow:hidden;padding:2px 8px;box-sizing:border-box;} Order By: Relevance .css-19bb58m{visibility:visible;-webkit-flex:1 1 auto;-ms-flex:1 1 auto;flex:1 1 auto;display:inline-grid;grid-area:1/1/2/3;grid-template-columns:0 min-content;margin:2px;padding-bottom:2px;padding-top:2px;color:hsl(0, 0%, 20%);box-sizing:border-box;}.css-19bb58m:after{content:attr(data-value) " ";visibility:hidden;white-space:pre;grid-area:1/2;font:inherit;min-width:2px;border:0;margin:0;outline:0;padding:0;} .css-1wy0on6{-webkit-align-items:center;-webkit-box-align:center;-ms-flex-align:center;align-items:center;-webkit-align-self:stretch;-ms-flex-item-align:stretch;align-self:stretch;display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-flex-shrink:0;-ms-flex-negative:0;flex-shrink:0;box-sizing:border-box;} .css-1xc3v61-indicatorContainer{display:-webkit-box;display:-webkit-flex;display:-ms-flexbox;display:flex;-webkit-transition:color 150ms;transition:color 150ms;color:hsl(0, 0%, 80%);padding:8px;box-sizing:border-box;}.css-1xc3v61-indicatorContainer:hover{color:hsl(0, 0%, 60%);} .css-8mmkcg{display:inline-block;fill:currentColor;line-height:1;stroke:currentColor;stroke-width:0;} Settings “…Klnç et al [8] proved Hadamard and Fejér-Hadamard inequalities for (h, m)-strongly convex functions via generalized fractional integrals with Mittag-Leffler functions. Jia et al [9] demonstrated new types of Hadamard and Fejér-Hadamard fractional integral inequalities for Riemann-Liouville fractional integrals. Yussouf et al [10] presented generalized types of Hadamard and Fejér-Hadamard-type fractional integral inequalities by utilizing generalized fractional integrals including Mittag-Leffler functions.…” Section : Introduction mentioning confidence: 99% (p, h)-Convex Functions Associated with Hadamard and Fejér-Hadamard Inequalities via k -Fractional Integral Operators Zhang 1, Farid 2, Demirel 3 et al. 2022 Journal of Function Spaces Self Cite 1000 View full text Add to dashboard Cite In this article, generalized versions of the

k

-fractional Hadamard and Fejér-Hadamard inequalities are constructed. To obtain the generalized versions of these inequalities,

k

-fractional integral operators including the well-known Mittag-Leffler function are utilized. The class of (

p
,
h

)-convex functions for Hadamard-type inequalities give the generalizations of results which have been proved in literature for

p

-convex,

h

-convex, and several functions deducible from these two classes. show abstract “…Klnç et al [8] proved Hadamard and Fejér-Hadamard inequalities for (h, m)-strongly convex functions via generalized fractional integrals with Mittag-Leffler functions. Jia et al [9] demonstrated new types of Hadamard and Fejér-Hadamard fractional integral inequalities for Riemann-Liouville fractional integrals. Yussouf et al [10] presented generalized types of Hadamard and Fejér-Hadamard-type fractional integral inequalities by utilizing generalized fractional integrals including Mittag-Leffler functions.…” Section : Introduction mentioning confidence: 99% (p, h)-Convex Functions Associated with Hadamard and Fejér-Hadamard Inequalities via k -Fractional Integral Operators Zhang 1, Farid 2, Demirel 3 et al. 2022 Journal of Function Spaces Self Cite 1000 View full text Add to dashboard Cite In this article, generalized versions of the

k

-fractional Hadamard and Fejér-Hadamard inequalities are constructed. To obtain the generalized versions of these inequalities,

k

-fractional integral operators including the well-known Mittag-Leffler function are utilized. The class of (

p
,
h

)-convex functions for Hadamard-type inequalities give the generalizations of results which have been proved in literature for

p

-convex,

h

-convex, and several functions deducible from these two classes. show abstract Some Generalized Formulas of Hadamard-Type Fractional Integral Inequalities Zhang 1, Farid 2, Yasmeen 3 et al. 2022 Journal of Function Spaces 1000 View full text Add to dashboard Cite This paper is aimed at establishing the generalized forms of Riemann-Liouville fractional inequalities of the Hadamard type for a class of functions known as strongly exponentially

α
,
h
−
m

-

p

-convex functions. These inequalities provide some general formulas from which one can get associated inequalities for various types of exponentially convex and strongly convex functions. Refinements of well-known inequalities are also deducible from the established theorems. show abstract New Generalized Riemann–Liouville Fractional Integral Versions of Hadamard and Fejér–Hadamard Inequalities Nonlaopon 1, Farid 2, Nosheen 3 et al. 2022 Journal of Mathematics 2020 View full text Add to dashboard Cite In this paper, a new class of functions, namely, exponentially

α
,
h
−
m

−
p

-convex functions is introduced to unify various classes of functions already defined in the subject of convex analysis. By using this class of functions, generalized versions of well known fractional integral inequalities of Hadamard and Fejér–Hadamard type are obtained. The results of this paper generate fractional integral inequalities of Hadamard and Fejér–Hadamard type for various types of convex and exponentially convex functions simultaneously. show abstractscite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health. Contact Info customersupport@researchsolutions.com 10624 S. Eastern Ave., Ste. A-614 Henderson, NV 89052, USA Product Browser Extension Assistant by scite Citation Statement Search Reference Check Visualizations Dashboards Explore Journals Explore Organizations Explore Funders Embedding Badge Embedding Citation Search Pricing Resources Blog Help & FAQ Accessibility Statement API Terms For Universities & Governments For Researchers For Publishers For Corporate, Pharma & Enterprise Author Marketing Become an Affiliate Get an organization trial or quote scite Data & Services About News & Press Careers Read our Paper Coverage This site is protected by reCAPTCHA and the Google Privacy Policy and Terms of Service apply. Blog Terms and Conditions API Terms Privacy Policy Contact Cookie Preferences Do Not Sell or Share My Personal Information Copyright © 2025 scite LLC. All rights reserved. Made with 💙 for researchers Part of the Research Solutions Family.window.__SCITE_STORE_STATE = {"addReportDashboardModal":{"selectedPapers":[]},"ui":{"referralPartner":null,"freeReports":0,"bannerContent":null,"bannerClosed":false,"navMenu":{},"createDashboardFromSearchResultsModalShow":false,"savedSearchModalShow":false,"searchFilterModalShow":false,"flagModalShow":false,"resetPasswordModalShow":false,"loginModalShow":false,"referenceCheckModalShow":false,"deleteDashboardModalShow":false,"addReportToDashboardModalShow":false,"shareDashboardModalShow":false,"editDashboardModalShow":false,"upgradeModalShow":false,"exportReportModalShow":false,"exportReferenceModalShow":false,"generateCitationModalShow":false,"exportDashboardModalShow":false,"dashboardLimitErrorModalShow":false,"whatIsThisReportModalShow":false,"whatIsThisSearchModalShow":false,"whatIsAssistantModalShow":false,"trialOverModalShow":false,"activateTrialModalShow":false,"cancelTrialCheckoutModalShow":false,"cookieModalShow":false,"personaModalShow":false,"loggedOutModalShow":true,"modalFromContext":"","shownModals":[],"hasUser":false,"user":null,"ipLocationData":{"data":null,"error":false,"loading":false},"fetchingUser":false,"savingUser":false,"saveUserError":false,"onboardingLoadTracker":{"state":undefined},"userEntitlements":null,"entitlementsLoading":false,"referralsLoading":false,"pageError":null,"browser":null,"extensionLoaded":false,"organizationLicense":{"hasLicense":false},"linkResolver":null,"sickoMode":false,"authors":[],"navOpen":false,"showMobileSearch":false,"redeemCodeLoading":false,"redeemCodeError":null,"redeemCodeSuccess":false,"applyDiscountLoading":false,"applyDiscountError":null,"applyDiscountSuccess":false,"userDiscount":null,"search":{"term":"","focused":false,"autocompletesLoadTracker":{},"autocompletes":{}},"referencesToExport":[],"citeGeneratorPaperData":{},"dimensions":{},"cookiePreference":null,"userChurnKey":null,"churnKeyLoading":false,"fullTextLinkState":{},"agAvailabilityState":{},"agAvailabilityQueue":[]},"verifyEmail":{"loading":false,"verified":false,"expired":false},"dashboard":{"id":"","name":"","displayName":"","normalizedName":"","description":"","slug":"","dashboardType":"","accessType":"","isPublic":null,"componentLayout":[],"doiQuery":{},"journalMetadata":{},"affiliationMetadata":{},"institutionMetadata":{},"haveAccessToDashboard":null,"lastUpdated":null,"doisUpdatedTS":null,"doisRefreshedTS":null,"refreshStatus":"","loadState":{},"deleteDashboardError":false,"deletingDashboard":false,"totalDoiCount":0,"matchedDoiCount":0,"unmatchedDoiCount":0,"exportLoadTracker":{},"setDashboardUsersLoading":false,"setDashboardUsersError":false,"setDashboardUsersSuccess":false,"dashboardAccessList":null,"getDashboardAccessListLoading":false,"getDashboardAccessListSuccess":false,"getDashboardAccessListError":false},"notifications":{"notifications":[],"loading":false,"error":null,"notificationItems":[],"notificationItemsLoading":false,"notificationItemsError":null,"notificationItemsPage":1,"notificationItemsTotal":0,"newNotificationItems":0,"notificationItemsFilters":{},"dropDownNotificationItems":[],"dropDownNotificationItemsLoading":false,"dropDownNotificationItemsError":null,"notificationSetStatusLoading":false,"notificationSetStatusError":null},"report":{"paper":"10.1155\u002F2021\u002F9945114","tally":"10.1155\u002F2021\u002F9945114","citations":[{"id":3459338750,"source":"10.1155\u002F2022\u002F3832330","target":"10.1155\u002F2021\u002F9945114","negative":0.017786124721169472,"positive":0.030856508761644363,"neutral":1,"section":"introduction","type":"mentioning","typeConfidence":1,"snippet":"Klnç et al \u003Ccite data-doi=\"10.33773\u002Fjum.506507\"\u003E[8]\u003C\u002Fcite\u003E proved Hadamard and Fejér-Hadamard inequalities for (h, m)-strongly convex functions via generalized fractional integrals with Mittag-Leffler functions. Jia et al \u003Ccite data-doi=\"10.1155\u002F2021\u002F9945114\"\u003E[9]\u003C\u002Fcite\u003E demonstrated new types of Hadamard and Fejér-Hadamard fractional integral inequalities for Riemann-Liouville fractional integrals. Yussouf et al \u003Ccite data-doi=\"10.1155\u002F2021\u002F5589405\"\u003E[10]\u003C\u002Fcite\u003E presented generalized types of Hadamard and Fejér-Hadamard-type fractional integral inequalities by utilizing generalized fractional integrals including Mittag-Leffler functions.","lang":"en","langConfidence":"0.7599999904632568","refLocation":"b8\u002F1","memberId":98,"selfCites":[{"type":"SHAREDAUTHOR","family":"Farid","given":"Ghulam"}],"snippetHidden":false},{"source":"10.1155\u002F2022\u002F3723867","target":"10.1155\u002F2021\u002F9945114","selfCites":[],"dataSource":"OPENALEX"},{"source":"10.1155\u002F2022\u002F8173785","target":"10.1155\u002F2021\u002F9945114","selfCites":[],"dataSource":"CR"},{"source":"10.1155\u002F2024\u002F4082683","target":"10.1155\u002F2021\u002F9945114","selfCites":[],"dataSource":"CR"},{"source":"10.3390\u002Faxioms12070719","target":"10.1155\u002F2021\u002F9945114","selfCites":[],"dataSource":"CR"}],"citationTallies":{"10.1155\u002F2022\u002F3832330":{"total":0,"supporting":0,"contradicting":0,"mentioning":0,"unclassified":0,"doi":"10.1155\u002F2022\u002F3832330","citingPublications":1},"10.1155\u002F2022\u002F3723867":{"total":0,"supporting":0,"contradicting":0,"mentioning":0,"unclassified":0,"doi":"10.1155\u002F2022\u002F3723867","citingPublications":1},"10.1155\u002F2022\u002F8173785":{"total":2,"supporting":0,"contradicting":0,"mentioning":2,"unclassified":0,"doi":"10.1155\u002F2022\u002F8173785","citingPublications":2},"10.1155\u002F2024\u002F4082683":{"total":0,"supporting":0,"contradicting":0,"mentioning":0,"unclassified":0,"doi":"10.1155\u002F2024\u002F4082683","citingPublications":0},"10.3390\u002Faxioms12070719":{"total":0,"supporting":0,"contradicting":0,"mentioning":0,"unclassified":0,"doi":"10.3390\u002Faxioms12070719","citingPublications":0}},"metadata":{"totalCitationCount":1,"restrictedCitationCount":5,"distinctSourceCount":13},"papers":{"10.1155\u002F2022\u002F3832330":{"id":11369586250,"doi":"10.1155\u002F2022\u002F3832330","slug":"math-xmlns-http-www-w3-org-1998-math-mathml-id-m1-mi-p-mi-mo-mo-zRg4EJwK","type":"journal-article","title":"(\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" id=\"M1\"\u003E\n \u003Cmi\u003Ep\u003C\u002Fmi\u003E\n \u003Cmo\u003E,\u003C\u002Fmo\u003E\n \u003Cmi\u003Eh\u003C\u002Fmi\u003E\n \u003C\u002Fmath\u003E)-Convex Functions Associated with Hadamard and Fejér-Hadamard Inequalities via \u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" id=\"M2\"\u003E\n \u003Cmi\u003Ek\u003C\u002Fmi\u003E\n \u003C\u002Fmath\u003E-Fractional Integral Operators","abstract":"In this article, generalized versions of the \n \n k\n \n -fractional Hadamard and Fejér-Hadamard inequalities are constructed. To obtain the generalized versions of these inequalities, \n \n k\n \n -fractional integral operators including the well-known Mittag-Leffler function are utilized. The class of (\n \n p\n ,\n h\n \n )-convex functions for Hadamard-type inequalities give the generalizations of results which have been proved in literature for \n \n p\n \n -convex, \n \n h\n \n -convex, and several functions deducible from these two classes.","authors":[{"family":"Zhang","given":"Xiujun","affiliation":"Chengdu University","authorSlug":"xiujun-zhang-kZY4gJ","authorName":"Xiujun Zhang","authorID":"8006268","authorLastKnownAffiliationId":162339,"authorSequenceNumber":1,"affiliationSlug":"chengdu-university-aXMeK","affiliationID":"162339"},{"family":"Farid","given":"Ghulam","affiliation":"COMSATS University Islamabad","authorSlug":"ghulam-farid-5Gzy6p","authorName":"Ghulam Farid","authorID":"6557569","authorLastKnownAffiliationId":13190,"authorSequenceNumber":2,"affiliationSlug":"comsats-university-islamabad-9N5G","affiliationID":"13190"},{"family":"Demirel","given":"Ayşe Kübra","affiliation":"Ordu University","authorSlug":"ayse-kubra-demirel-04e0mZ","authorName":"Ayşe Kübra Demirel","authorID":"23898856","authorLastKnownAffiliationId":15244,"authorSequenceNumber":3,"affiliationSlug":"ordu-university-3zRd","affiliationID":"15244"},{"family":"Jung","given":"Chahn Yong","affiliation":"Gyeongsang National University","authorSlug":"chahn-yong-jung-Xe5bgE","authorName":"Chahn Yong Jung","authorID":"10772085","authorLastKnownAffiliationId":2442,"authorSequenceNumber":4,"affiliationSlug":"gyeongsang-national-university-4LOR","affiliationID":"2442"}],"keywords":[],"year":2022,"shortJournal":"Journal of Function Spaces","publisher":"Hindawi Limited","volume":"2022","page":"1-12","memberId":98,"issns":["2314-8888","2314-8896"],"editorialNotices":[],"journalSlug":"journal-of-function-spaces-EWmL1","journal":"Journal of Function Spaces","preprintLinks":[],"publicationLinks":[],"normalizedTypes":["article"]},"10.3390\u002Faxioms12070719":{"id":11851009400,"doi":"10.3390\u002Faxioms12070719","slug":"a-comprehensive-review-on-the-3nx9RLmW","type":"journal-article","title":"A Comprehensive Review on the Fejér-Type Inequality Pertaining to Fractional Integral Operators","abstract":"A review of the results on the fractional Fejér-type inequalities, associated with different families of convexities and different kinds of fractional integrals, is presented. In the numerous families of convexities, it includes classical convex functions, s-convex functions, quasi-convex functions, strongly convex functions, harmonically convex functions, harmonically quasi-convex functions, quasi-geometrically convex functions, p-convex functions, convexity with respect to strictly monotone function, co-ordinated-convex functions, (θ,h−m)−p-convex functions, and h-preinvex functions. Included in the fractional integral operators are Riemann–Liouville fractional integral, (k−p)-Riemann–Liouville, k-Riemann–Liouville fractional integral, Riemann–Liouville fractional integrals with respect to another function, the weighted fractional integrals of a function with respect to another function, fractional integral operators with the exponential kernel, Hadamard fractional integral, Raina fractional integral operator, conformable integrals, non-conformable fractional integral, and Katugampola fractional integral. Finally, Fejér-type fractional integral inequalities for invex functions and (p,q)-calculus are also included.","authors":[{"family":"Tariq","given":"Muhammad","affiliation":"Mehran University of Engineering and Technology","authorSlug":"muhammad-tariq-QedeZM","authorName":"Muhammad Tariq","authorID":"6988760","authorLastKnownAffiliationId":23896,"authorSequenceNumber":1,"affiliationSlug":"mehran-university-of-engineering-and-M9Gl","affiliationID":"23896"},{"family":"Ntouyas","given":"Sotiris K.","affiliation":"University of Ioannina","authorSlug":"sotiris-k-ntouyas-vJ0pP2","authorName":"Sotiris K. Ntouyas","authorID":"5979109","authorLastKnownAffiliationId":154,"authorSequenceNumber":2,"affiliationSlug":"university-of-ioannina-6nk","affiliationID":"154"},{"family":"Shaikh","given":"Asif Ali","affiliation":"Mehran University of Engineering and Technology","authorSlug":"asif-ali-shaikh-lXZl4M","authorName":"Asif Ali Shaikh","authorID":"7623836","authorLastKnownAffiliationId":12475,"authorSequenceNumber":3,"affiliationSlug":"mehran-university-of-engineering-and-M9Gl","affiliationID":"23896"}],"keywords":["Tariq, M.","Ntouyas, S.K.","Shaikh, A.A. A Comprehensive Axioms 2023, 12, 719 Fejér inequality","Hadamard fractional integral","conformable and non-conformable fractional integral","Katugampola fractional integral","Riemann-Liouville fractional integral","(p, q)-calculus MSC: 26A33","26A51","26D07","26D10","26D15"],"year":2023,"shortJournal":"Axioms","publisher":"MDPI AG","issue":"7","volume":"12","page":"719","memberId":1968,"issns":["2075-1680"],"editorialNotices":[],"journalSlug":"axioms-NlA3a","journal":"Axioms","preprintLinks":[],"publicationLinks":[],"normalizedTypes":["article"]},"10.1155\u002F2022\u002F3723867":{"id":11723469272,"doi":"10.1155\u002F2022\u002F3723867","slug":"some-generalized-formulas-of-hadamard-type-wmvZnwJz","type":"journal-article","title":"Some Generalized Formulas of Hadamard-Type Fractional Integral Inequalities","abstract":"This paper is aimed at establishing the generalized forms of Riemann-Liouville fractional inequalities of the Hadamard type for a class of functions known as strongly exponentially \n \n \n \n α\n ,\n h\n −\n m\n \n \n \n -\n \n p\n \n -convex functions. These inequalities provide some general formulas from which one can get associated inequalities for various types of exponentially convex and strongly convex functions. Refinements of well-known inequalities are also deducible from the established theorems.","authors":[{"family":"Zhang","given":"Xiujun","affiliation":"Chengdu University","authorSlug":"xiujun-zhang-kZY4gJ","authorName":"Xiujun Zhang","authorID":"8006268","authorLastKnownAffiliationId":162339,"authorSequenceNumber":1,"affiliationSlug":"chengdu-university-aXMeK","affiliationID":"162339"},{"family":"Farid","given":"Ghulam","affiliation":"COMSATS University Islamabad","authorSlug":"ghulam-farid-5Gzy6p","authorName":"Ghulam Farid","authorID":"6557569","authorLastKnownAffiliationId":13190,"authorSequenceNumber":2,"affiliationSlug":"comsats-university-islamabad-9N5G","affiliationID":"13190"},{"family":"Yasmeen","given":"Hafsa","affiliation":"COMSATS University Islamabad","authorSlug":"hafsa-yasmeen-zzPllQ","authorName":"Hafsa Yasmeen","authorID":"16002694","authorLastKnownAffiliationId":13190,"authorSequenceNumber":3,"affiliationSlug":"comsats-university-islamabad-9N5G","affiliationID":"13190"},{"family":"Nonlaopon","given":"Kamsing","affiliation":"Khon Kaen University","authorSlug":"kamsing-nonlaopon-2NZ2EZ","authorName":"Kamsing Nonlaopon","authorID":"7197032","authorLastKnownAffiliationId":1788,"authorSequenceNumber":4,"affiliationSlug":"khon-kaen-university-32J","affiliationID":"1788"}],"year":2022,"shortJournal":"Journal of Function Spaces","publisher":"Hindawi Limited","volume":"2022","page":"1-12","memberId":98,"issns":["2314-8888","2314-8896"],"editorialNotices":[],"journalSlug":"journal-of-function-spaces-EWmL1","journal":"Journal of Function Spaces","preprintLinks":[],"publicationLinks":[],"normalizedTypes":["article"]},"10.1155\u002F2022\u002F8173785":{"id":11377825919,"doi":"10.1155\u002F2022\u002F8173785","slug":"new-generalized-riemann-liouville-fractional-integral-D1MYwXaN","type":"journal-article","title":"New Generalized Riemann–Liouville Fractional Integral Versions of Hadamard and Fejér–Hadamard Inequalities","abstract":"In this paper, a new class of functions, namely, exponentially \n \n \n \n α\n ,\n h\n −\n m\n \n \n −\n p\n \n -convex functions is introduced to unify various classes of functions already defined in the subject of convex analysis. By using this class of functions, generalized versions of well known fractional integral inequalities of Hadamard and Fejér–Hadamard type are obtained. The results of this paper generate fractional integral inequalities of Hadamard and Fejér–Hadamard type for various types of convex and exponentially convex functions simultaneously.","authors":[{"family":"Nonlaopon","given":"Kamsing","affiliation":"Khon Kaen University","authorSlug":"kamsing-nonlaopon-2NZ2EZ","authorName":"Kamsing Nonlaopon","authorID":"7197032","authorLastKnownAffiliationId":1788,"authorSequenceNumber":1,"affiliationSlug":"khon-kaen-university-32J","affiliationID":"1788"},{"family":"Farid","given":"Ghulam","affiliation":"COMSATS University Islamabad","authorSlug":"ghulam-farid-5Gzy6p","authorName":"Ghulam Farid","authorID":"6557569","authorLastKnownAffiliationId":13190,"authorSequenceNumber":2,"affiliationSlug":"comsats-university-islamabad-9N5G","affiliationID":"13190"},{"family":"Nosheen","given":"Ammara","affiliation":"University of Sargodha","authorSlug":"ammara-nosheen-PzMAPg","authorName":"Ammara Nosheen","authorID":"10161545","authorLastKnownAffiliationId":7794,"authorSequenceNumber":3,"affiliationSlug":"university-of-sargodha-zzRx","affiliationID":"7794"},{"family":"Yussouf","given":"Muhammad","affiliation":"University of Sargodha","authorSlug":"muhammad-yussouf-1n5prM","authorName":"Muhammad Yussouf","authorID":"11748647","authorLastKnownAffiliationId":7794,"authorSequenceNumber":4,"affiliationSlug":"university-of-sargodha-zzRx","affiliationID":"7794"},{"family":"Bonyah","given":"Ebenezer","authorSlug":"ebenezer-bonyah-gZ5WE9","authorName":"Ebenezer Bonyah","authorID":"6723458","authorLastKnownAffiliationId":2562,"authorSequenceNumber":5}],"year":2022,"shortJournal":"Journal of Mathematics","publisher":"Wiley","issue":"1","volume":"2022","memberId":311,"issns":["2314-4629","2314-4785"],"editorialNotices":[],"journalSlug":"journal-of-mathematics-W858Q","journal":"Journal of Mathematics","preprintLinks":[],"publicationLinks":[],"normalizedTypes":["article"]},"10.1155\u002F2024\u002F4082683":{"id":12341285032,"doi":"10.1155\u002F2024\u002F4082683","slug":"refinements-of-various-types-of-2NVnpjV1","type":"journal-article","title":"Refinements of Various Types of Fractional Inequalities via Generalized Convexity","abstract":"This paper aims to find generalizations of inequalities that hold for unified integral operators by applying strongly exponentially (α, ℏ − m) − p‐convex functions. These inequalities generate results for several fractional integral operators and simultaneously hold for convex, strongly convex, and exponentially convex functions. The particular cases of presented inequalities are directly connected with many recently published results.","authors":[{"family":"Tang","given":"Yong"},{"family":"Farid","given":"Ghulam","orcid":"0000-0002-4103-7745"},{"family":"Youssif","given":"M. Y."},{"family":"Aboabuda","given":"Zakieldeen"},{"family":"Elhag","given":"Amna E."},{"family":"Mahreen","given":"Kahkashan"}],"year":2024,"shortJournal":"Journal of Mathematics","publisher":"Wiley","issue":"1","volume":"2024","memberId":311,"issns":["2314-4629","2314-4785"],"editorialNotices":[],"journalSlug":"journal-of-mathematics-W858Q","journal":"Journal of Mathematics","preprintLinks":[],"publicationLinks":[],"normalizedTypes":["article"]},"10.1155\u002F2021\u002F9945114":{"id":11158815880,"doi":"10.1155\u002F2021\u002F9945114","slug":"hadamard-and-fejer-hadamard-inequalities-for-lZn5m2Oa","type":"journal-article","title":"Hadamard and Fejér–Hadamard Inequalities for \u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" id=\"M1\"\u003E\n \u003Cmfenced open=\"(\" close=\")\" separators=\"|\"\u003E\n \u003Cmrow\u003E\n \u003Cmi\u003Eα\u003C\u002Fmi\u003E\n \u003Cmo\u003E,\u003C\u002Fmo\u003E\n \u003Cmi\u003Eh\u003C\u002Fmi\u003E\n \u003Cmo\u003E−\u003C\u002Fmo\u003E\n \u003Cmi\u003Em\u003C\u002Fmi\u003E\n \u003C\u002Fmrow\u003E\n \u003C\u002Fmfenced\u003E\n \u003Cmo\u003E−\u003C\u002Fmo\u003E\n \u003Cmi\u003Ep","abstract":"In this paper, we introduce \n \n \n \n α\n ,\n h\n −\n m\n \n \n −\n p\n \n -convex function and some related functions. By applying this generalized definition, new versions of Hadamard and Fejér–Hadamard fractional integral inequalities for Riemann–Liouville fractional integrals are given. The presented results hold at the same time for different types of convexities.","authors":[{"family":"Jia","given":"Wenyan","affiliation":"Sias University","authorSlug":"wenyan-jia-mO5ZZg","authorName":"Wenyan Jia","authorID":"6200587","authorLastKnownAffiliationId":16101,"authorSequenceNumber":1,"affiliationSlug":"sias-university-G5a1","affiliationID":"4157"},{"family":"Yussouf","given":"Muhammad","affiliation":"University of Sargodha","authorSlug":"muhammad-yussouf-1n5prM","authorName":"Muhammad Yussouf","authorID":"11748647","authorLastKnownAffiliationId":7794,"authorSequenceNumber":2,"affiliationSlug":"university-of-sargodha-zzRx","affiliationID":"7794"},{"family":"Farid","given":"Ghulam","affiliation":"COMSATS University Islamabad","authorSlug":"ghulam-farid-5Gzy6p","authorName":"Ghulam Farid","authorID":"6557569","authorLastKnownAffiliationId":13190,"authorSequenceNumber":3,"affiliationSlug":"comsats-university-islamabad-9N5G","affiliationID":"13190"},{"family":"Khan","given":"Khuram Ali","affiliation":"University of Sargodha","authorSlug":"khuram-ali-khan-24L8x","authorName":"Khuram Ali Khan","authorID":"1278976","authorLastKnownAffiliationId":7794,"authorSequenceNumber":4,"affiliationSlug":"university-of-sargodha-zzRx","affiliationID":"7794"}],"keywords":[],"year":2021,"shortJournal":"Mathematical Problems in Engineering","publisher":"Hindawi Limited","volume":"2021","page":"1-12","memberId":98,"issns":["1563-5147","1024-123X"],"editorialNotices":[],"journalSlug":"mathematical-problems-in-engineering-0G2v9","journal":"Mathematical Problems in Engineering","preprintLinks":[],"publicationLinks":[],"normalizedTypes":["article"]}},"tallies":{"10.1155\u002F2021\u002F9945114":{"total":1,"supporting":0,"contradicting":0,"mentioning":1,"unclassified":0,"doi":"10.1155\u002F2021\u002F9945114","citingPublications":13}},"selectCitation":null,"references":[],"referencesTotalCount":19,"referencePapers":{},"referenceTallies":{},"referencesByPapers":[],"paperLoadTracker":{"state":"loaded","key":"hadamard-and-fejer-hadamard-inequalities-for-lZn5m2Oa"},"citingCountsLoadTracker":{"state":"loading"},"citationsLoadTracker":{"state":"loaded"},"referencesLoadTracker":{"state":"loaded"},"claimPaperLoadTracker":{},"flagCitationLoadTracker":{},"filters":{"showMobile":false,"sort":{"id":"relevancy","value":null,"label":"Relevance"},"supporting":null,"mentioning":null,"contradicting":null,"unclassified":null,"selfCite":null,"independent":null,"publishedRange":null,"citationsPerPage":10,"currentPage":1,"textMatch":"","citationIDs":[],"sourceDOIIDs":[],"withCitationStatements":true,"withoutCitationStatements":true,"journals":[],"authors":[],"pubtypes":[],"sections":[],"showReferences":false},"flag":{"citation":null,"citationType":null,"type":"","comment":""},"moreLikeThisResults":null,"moreLikeThisLoading":false,"exportReportLoading":false},"loginModal":{"mode":"login","name":{"placeholder":"Full Name","type":"text","value":""},"email":{"type":"email","placeholder":"Email","value":""},"password":{"type":"password","placeholder":"Password","value":""},"repeatPassword":{"type":"password","placeholder":"Repeat Password","value":""},"privacyAgreed":false,"loading":false,"userSignupType":null,"userSignupTypeLoading":false,"userSignupTypeEmail":null,"userSignupTypeOrg":null,"userSignupTypeEmailDomains":null,"fetchedSignupType":false,"fetchedSignupTypeError":false}}[]{"namedChunks":[]}(function(i,s,o,g,r,a,m){i[o]=i[o]||function(){(i[o].q=i[o].q||[]).push(arguments)};
a=s.createElement(g);m=s.getElementsByTagName(g)[0];a.async=1;a.src=r+'?auth='+
s.getElementById(o+'-js').getAttribute('data-pw-auth');m.parentNode.insertBefore(a,m);
})(window,document,'profitwell','script','https://public.profitwell.com/js/profitwell.js'); window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag("consent", "default", {ad_storage: "denied", analytics_storage: "denied", wait_for_update: 1500}); gtag('js', new Date()); gtag('config', 'AW-614125746'); (function(t,a,p){t.TapfiliateObject=a;t[a]=t[a]||function(){ (t[a].q=t[a].q||[]).push(arguments)}})(window,'tap');

tap('create', '48731-1a14bc', { integration: "javascript" });
tap('detect'); window.__MathJax_State = {
isReady: false,
promise: new Promise(resolve => {
window.MathJax = {
// MathJax can be configured as desired in addition to these options.
startup: {
ready: () => {
// Do whatever MathJax would normally do at this point.
MathJax.startup.defaultReady();
// Set the flag and resolve the promise.
window.MathJax_State__.isReady = true;
resolve();
}
},
tex: {
inlineMath: [
["$", "$"],
],
},
svg: {
fontCache: "global"
}
};
})
}; var _rollbarConfig = {
accessToken: "2c3f1a7fb80c4667a8370b6d5e60b850",
captureUncaught: true,
captureUnhandledRejections: true,
payload: {
environment: "staging",
client: {
javascript: {
code_version: "null"
}
}
}
};
// Rollbar Snippet
!function(r){function e(n){if(o[n])return o[n].exports;var t=o[n]={exports:{},id:n,loaded:!1};return r[n].call(t.exports,t,t.exports,e),t.loaded=!0,t.exports}var o={};return e.m=r,e.c=o,e.p="",e(0)}([function(r,e,o){"use strict";var n=o(1),t=o(4);_rollbarConfig=_rollbarConfig||{},_rollbarConfig.rollbarJsUrl=_rollbarConfig.rollbarJsUrl||"https://cdnjs.cloudflare.com/ajax/libs/rollbar.js/2.24.0/rollbar.min.js",_rollbarConfig.async=void 0===_rollbarConfig.async||_rollbarConfig.async;var a=n.setupShim(window,_rollbarConfig),l=t(_rollbarConfig);window.rollbar=n.Rollbar,a.loadFull(window,document,!_rollbarConfig.async,_rollbarConfig,l)},function(r,e,o){"use strict";function n(r){return function(){try{return r.apply(this,arguments)}catch(r){try{console.error("[Rollbar]: Internal error",r)}catch(r){}}}}function t(r,e){this.options=r,this._rollbarOldOnError=null;var o=s++;this.shimId=function(){return o},"undefined"!=typeof window&&window._rollbarShims&&(window._rollbarShims[o]={handler:e,messages:[]})}function a(r,e){if(r){var o=e.globalAlias||"Rollbar";if("object"==typeof r[o])return r[o];r._rollbarShims={},r._rollbarWrappedError=null;var t=new p(e);return n(function(){e.captureUncaught&&(t._rollbarOldOnError=r.onerror,i.captureUncaughtExceptions(r,t,!0),i.wrapGlobals(r,t,!0)),e.captureUnhandledRejections&&i.captureUnhandledRejections(r,t,!0);var n=e.autoInstrument;return e.enabled!==!1&&(void 0===n||n===!0||"object"==typeof n&&n.network)&&r.addEventListener&&(r.addEventListener("load",t.captureLoad.bind(t)),r.addEventListener("DOMContentLoaded",t.captureDomContentLoaded.bind(t))),r[o]=t,t})()}}function l(r){return n(function(){var e=this,o=Array.prototype.slice.call(arguments,0),n={shim:e,method:r,args:o,ts:new Date};window._rollbarShims[this.shimId()].messages.push(n)})}var i=o(2),s=0,d=o(3),c=function(r,e){return new t(r,e)},p=d.bind(null,c);t.prototype.loadFull=function(r,e,o,t,a){var l=function(){var e;if(void 0===r._rollbarDidLoad){e=new Error("rollbar.js did not load");for(var o,n,t,l,i=0;o=r._rollbarShims[i++];)for(o=o.messages||[];n=o.shift();)for(t=n.args||[],i=0;i<t.length;++i)if(l=t[i],"function"==typeof l){l(e);break}}"function"==typeof a&&a(e)},i=!1,s=e.createElement("script"),d=e.getElementsByTagName("script")[0],c=d.parentNode;s.crossOrigin="",s.src=t.rollbarJsUrl,o||(s.async=!0),s.onload=s.onreadystatechange=n(function(){if(!(i||this.readyState&&"loaded"!==this.readyState&&"complete"!==this.readyState)){s.onload=s.onreadystatechange=null;try{c.removeChild(s)}catch(r){}i=!0,l()}}),c.insertBefore(s,d)},t.prototype.wrap=function(r,e,o){try{var n;if(n="function"==typeof e?e:function(){return e||{}},"function"!=typeof r)return r;if(r._isWrap)return r;if(!r._rollbar_wrapped&&(r._rollbar_wrapped=function(){o&&"function"==typeof o&&o.apply(this,arguments);try{return r.apply(this,arguments)}catch(o){var e=o;throw e&&("string"==typeof e&&(e=new String(e)),e._rollbarContext=n()||{},e._rollbarContext._wrappedSource=r.toString(),window._rollbarWrappedError=e),e}},r._rollbar_wrapped._isWrap=!0,r.hasOwnProperty))for(var t in r)r.hasOwnProperty(t)&&(r._rollbar_wrapped[t]=r[t]);return r._rollbar_wrapped}catch(e){return r}};for(var u="log,debug,info,warn,warning,error,critical,global,configure,handleUncaughtException,handleUnhandledRejection,captureEvent,captureDomContentLoaded,captureLoad".split(","),f=0;f<u.length;++f)t.prototype[u[f]]=l(u[f]);r.exports={setupShim:a,Rollbar:p}},function(r,e){"use strict";function o(r,e,o){if(r){var t;"function"==typeof e._rollbarOldOnError?t=e._rollbarOldOnError:r.onerror&&!r.onerror.belongsToShim&&(t=r.onerror,e._rollbarOldOnError=t);var a=function(){var o=Array.prototype.slice.call(arguments,0);n(r,e,t,o)};a.belongsToShim=o,r.onerror=a}}function n(r,e,o,n){r._rollbarWrappedError&&(n[4]||(n[4]=r._rollbarWrappedError),n[5]||(n[5]=r._rollbarWrappedError._rollbarContext),r._rollbarWrappedError=null),e.handleUncaughtException.apply(e,n),o&&o.apply(r,n)}function t(r,e,o){if(r){"function"==typeof r._rollbarURH&&r._rollbarURH.belongsToShim&&r.removeEventListener("unhandledrejection",r._rollbarURH);var n=function(r){var o,n,t;try{o=r.reason}catch(r){o=void 0}try{n=r.promise}catch(r){n="[unhandledrejection] error getting `promise` from event"}try{t=r.detail,!o&&t&&(o=t.reason,n=t.promise)}catch(r){t="[unhandledrejection] error getting `detail` from event"}o||(o="[unhandledrejection] error getting `reason` from event"),e&&e.handleUnhandledRejection&&e.handleUnhandledRejection(o,n)};n.belongsToShim=o,r._rollbarURH=n,r.addEventListener("unhandledrejection",n)}}function a(r,e,o){if(r){var n,t,a="EventTarget,Window,Node,ApplicationCache,AudioTrackList,ChannelMergerNode,CryptoOperation,EventSource,FileReader,HTMLUnknownElement,IDBDatabase,IDBRequest,IDBTransaction,KeyOperation,MediaController,MessagePort,ModalWindow,Notification,SVGElementInstance,Screen,TextTrack,TextTrackCue,TextTrackList,WebSocket,WebSocketWorker,Worker,XMLHttpRequest,XMLHttpRequestEventTarget,XMLHttpRequestUpload".split(",");for(n=0;n<a.length;++n)t=a[n],r[t]&&r[t].prototype&&l(e,r[t].prototype,o)}}function l(r,e,o){if(e.hasOwnProperty&&e.hasOwnProperty("addEventListener")){for(var n=e.addEventListener;n._rollbarOldAdd&&n.belongsToShim;)n=n._rollbarOldAdd;var t=function(e,o,t){n.call(this,e,r.wrap(o),t)};t._rollbarOldAdd=n,t.belongsToShim=o,e.addEventListener=t;for(var a=e.removeEventListener;a._rollbarOldRemove&&a.belongsToShim;)a=a._rollbarOldRemove;var l=function(r,e,o){a.call(this,r,e&&e._rollbar_wrapped||e,o)};l._rollbarOldRemove=a,l.belongsToShim=o,e.removeEventListener=l}}r.exports={captureUncaughtExceptions:o,captureUnhandledRejections:t,wrapGlobals:a}},function(r,e){"use strict";function o(r,e){this.impl=r(e,this),this.options=e,n(o.prototype)}function n(r){for(var e=function(r){return function(){var e=Array.prototype.slice.call(arguments,0);if(this.impl[r])return this.impl[r].apply(this.impl,e)}},o="log,debug,info,warn,warning,error,critical,global,configure,handleUncaughtException,handleUnhandledRejection,_createItem,wrap,loadFull,shimId,captureEvent,captureDomContentLoaded,captureLoad".split(","),n=0;n<o.length;n++)r[o[n]]=e(o[n])}o.prototype._swapAndProcessMessages=function(r,e){this.impl=r(this.options);for(var o,n,t;o=e.shift();)n=o.method,t=o.args,this[n]&&"function"==typeof this[n]&&("captureDomContentLoaded"===n||"captureLoad"===n?this[n].apply(this,[t[0],o.ts]):this[n].apply(this,t));return this},r.exports=o},function(r,e){"use strict";r.exports=function(r){return function(e){if(!e&&!window._rollbarInitialized){r=r||{};for(var o,n,t=r.globalAlias||"Rollbar",a=window.rollbar,l=function(r){return new a(r)},i=0;o=window._rollbarShims[i++];)n||(n=o.handler),o.handler._swapAndProcessMessages(l,o.messages);window[t]=n,window._rollbarInitialized=!0}}}}]);
// End Rollbar Snippet (function(w,d,t,r,u)
{
var f,n,i;
w[u]=w[u]||[],f=function()
{
var o={ti:"343050743", enableAutoSpaTracking: true};
o.q=w[u],w[u]=new UET(o),w[u].push("pageLoad")
},
n=d.createElement(t),n.src=r,n.async=1,n.onload=n.onreadystatechange=function()
{
var s=this.readyState;
s&&s!=="loaded"&&s!=="complete"||(f(),n.onload=n.onreadystatechange=null)
},
i=d.getElementsByTagName(t)[0],i.parentNode.insertBefore(n,i)
})
(window,document,"script","//bat.bing.com/bat.js","uetq");!function(f,b,e,v,n,t,s)
{if(f.fbq)return;n=f.fbq=function(){n.callMethod?
n.callMethod.apply(n,arguments):n.queue.push(arguments)};
if(!f._fbq)f._fbq=n;n.push=n;n.loaded=!0;n.version='2.0';
n.queue=[];t=b.createElement(e);t.async=!0;
t.src=v;s=b.getElementsByTagName(e)[0];
s.parentNode.insertBefore(t,s)}(window, document,'script',
'https://connect.facebook.net/en_US/fbevents.js');
fbq('init', '927411631440706');
fbq('track', 'PageView'); <img height="1" width="1" style="display:none"
src="https://www.facebook.com/tr?id=927411631440706&ev=PageView&noscript=1"
/>!function(){
if (!window.churnkey || !window.churnkey.created) {
window.churnkey = { created: true };
const a = document.createElement('script');
a.src = 'https://assets.churnkey.co/js/app.js?appId=2l1x8e2dl';
a.async = true;
const b = document.getElementsByTagName('script')[0];
b.parentNode.insertBefore(a, b);
}
}(); <script src="https://tag.clearbitscripts.com/v1/pk_fc279c8e6fe02304f4b6e672bc239206/tags.js" referrerpolicy="strict-origin-when-cross-origin">$