Constantin’s inequality for nabla and diamond-alpha derivative

Güvenilir, Ayşe Feza; Kaymakçalan, Bıllûr; Pelen, Neslihan Nesliye

doi:10.1186/s13660-015-0681-9

Cited by 13 publications

(8 citation statements)

References 13 publications

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“…Lemma 2.6 (Chain rule for the nabla derivative) [16] If φ : R → R is continuously differentiable and…”

Section: Lemma 24 (Hölder's Inequality)mentioning

confidence: 99%

Hardy–Copson type inequalities for nabla time scale calculus

Kayar¹,

Kaymakçalan²

2021

Turk J Math

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This paper is devoted to the nabla unification of the discrete and continuous Hardy-Copson-type inequalities. Some of the obtained inequalities are nabla counterparts of their delta versions while the others are new even for the discrete, continuous, and delta cases. Moreover, these dynamic inequalities not only generalize and unify the related ones in the literature but also improve them in the special cases.

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“…Lemma 2.6 (Chain rule for the nabla derivative) [16] If φ : R → R is continuously differentiable and…”

Section: Lemma 24 (Hölder's Inequality)mentioning

confidence: 99%

Hardy–Copson type inequalities for nabla time scale calculus

Kayar¹,

Kaymakçalan²

2021

Turk J Math

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“…53 This calculus deals with diamond alpha denoted by ◊ 𝛼 differentiable and diamond alpha integrable functions which are convex linear combinations of the delta and the nabla differentiable and integrable functions, respectively. For some developments of this calculus and for some integral inequalities in this calculus, we refer to previous works 32,47,[54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69] and references therein.…”

Section: Preliminariesmentioning

confidence: 99%

“…[35][36][37][38][39][40] For some results about the nabla differential equations and nabla inequalities, see previous works. [41][42][43][44][45][46][47][48][49][50][51] Contrary to delta case, Bennett-Leindler type inequalities had not been considered until the paper 52 appeared. The nabla time scale unifications of the discrete Bennett-Leindler inequalities (9) and (10) and the continuous Bennett-Leindler inequalities ( 13) and ( 14) as well as the nabla versions of Theorems 1-4 for an arbitrary time scale can be seen in the next theorems.…”

Section: Introductionmentioning

confidence: 99%

Diamond alpha Bennett‐Leindler type dynamic inequalities and their applications

Kayar

Kaymakçalan

Pelen

2021

Math Methods in App Sciences

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In this paper, two kinds of dynamic Bennett-Leindler type inequalities via the diamond alpha integrals are derived. The first kind consists of eight new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together due to the fact that convex linear combinations of delta and nabla Bennett-Leindler type inequalities give diamond alpha Bennett-Leindler type inequalities. The second kind involves four new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Bennett-Leindler type inequalities. For the second type, choosing 𝛼 = 1 or 𝛼 = 0 not only yields the same results as the ones obtained for delta and nabla cases but also provides novel results for them.Therefore, both kinds of our results expand some of the known delta and nabla Bennett-Leindler type inequalities, offer new types of these inequalities, and bind and unify them into one diamond alpha Bennett-Leindler type inequalities.Moreover, an application of dynamic Bennett-Leindler type inequalities to the oscillation theory of the second-order half linear dynamic equation is developed and presented for the first time ever.

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“…The surveys [1,58] and the monograph [3] can be used to see these extended dynamic inequalities for delta approach. Although the nabla dynamic inequalities are less attractive compared to the delta ones, some of the nabla dynamic inequalities can be found in [6,13,27,40,41,53,56].…”

Section: Introductionmentioning

confidence: 99%

Diamond alpha Hardy-Copson type dynamic inequalities

Kayar

Kaymakçalan

2022

Hacettepe Journal of Mathematics and Statistics

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In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.

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Constantin’s inequality for nabla and diamond-alpha derivative

Cited by 13 publications

References 13 publications

Hardy–Copson type inequalities for nabla time scale calculus

Hardy–Copson type inequalities for nabla time scale calculus

Diamond alpha Bennett‐Leindler type dynamic inequalities and their applications

Diamond alpha Hardy-Copson type dynamic inequalities

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