Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means

Inequality theory provides a significant mechanism for managing symmetrical aspects in real-life circumstances. The renowned distinguishing feature of integral inequalities and fractional calculus has a solid possibility to regulate continuous issues with high proficiency. This manuscript contributes to a captivating association of fractional calculus, special functions and convex functions. The authors develop a novel approach for investigating a new class of convex functions which is known as an n-polynomial $\mathcal{P}$ P -convex function. Meanwhile, considering two identities via generalized fractional integrals, provide several generalizations of the Hermite–Hadamard and Ostrowski type inequalities by employing the better approaches of Hölder and power-mean inequalities. By this new strategy, using the concept of n-polynomial $\mathcal{P}$ P -convexity we can evaluate several other classes of n-polynomial harmonically convex, n-polynomial convex, classical harmonically convex and classical convex functions as particular cases. In order to investigate the efficiency and supremacy of the suggested scheme regarding the fractional calculus, special functions and n-polynomial $\mathcal{P}$ P -convexity, we present two applications for the modified Bessel function and $\mathfrak{q}$ q -digamma function. Finally, these outcomes can evaluate the possible symmetric roles of the criterion that express the real phenomena of the problem.

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“…Recalling the series representation of the first kind modified Bessel function [38] U ρ (z) = n≥0 (z/2) ρ+2n n! (ρ + n + 1) (z ∈ R).…”

Section: Inequalities For Special Function 61 Modified Bessel Functionmentioning

confidence: 99%

New fractional approaches for n-polynomial P-convexity with applications in special function theory

et al. 2020

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“…In the case of a + b = 1, these functions coincide with the special functions μ a (r), K a (r), E a (r), and m a (r), respectively, which were studied in [36][37][38][39][40][41][42][43][44][45][46][47][48][49]. In particular, if a = b = 1/2, then these functions reduce to the classical cases denoted by μ(r), K(r), E(r), and m(r), which appeared frequently in the geometric function theory and number theory [50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68][69].…”

Section: Introductionmentioning

confidence: 99%

Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function

Hai

Zhao

2020

J Inequal Appl

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“…Inequality plays an irreplaceable role in the development of mathematics. Very recently, many new inequalities such as Hermite-Hadamard type inequality [34][35][36][37][38], Petrović type inequality [39], Pólya-Szegö type inequality [40], Ostrowski type inequality [41], reverse Minkowski inequality [42], Jensen type inequality [43,44], Bessel function inequality [45], trigonometric and hyperbolic function inequalities [46], fractional integral inequality [47][48][49][50][51], complete and generalized elliptic integral inequalities [52][53][54][55][56][57], generalized convex function inequality [58][59][60], and mean value inequality [61][62][63] have been discovered by many researchers. In particular, the applications of integral inequalities have gained considerable importance among researchers for fixed-point theorems; the existence and uniqueness of solutions for differential equations [64][65][66][67][68] and numerous numerical and analytical methods have been recommended for the advancement of integral inequalities [69][70][71][72][73][74]…”

Section: Introductionmentioning

confidence: 99%

On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus

Rashid

Khalid

Rahman

et al. 2020

Journal of Function Spaces

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In the article, we present several generalizations for the generalized Čebyšev type inequality in the frame of quantum fractional Hahn’s integral operator by using the quantum shift operator σΨqς=qς+1−qσς∈l1,l2,σ=l1+ω/1−q,0<q<1,ω≥0. As applications, we provide some associated variants to illustrate the efficiency of quantum Hahn’s integral operator and compare our obtained results and proposed technique with the previously known results and existing technique. Our ideas and approaches may lead to new directions in fractional quantum calculus theory.

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Convexity and concavity of the modified Bessel functions of the first kind with respect to Hölder means

Cited by 101 publications

References 32 publications

New fractional approaches for n-polynomial P-convexity with applications in special function theory

New fractional approaches for n-polynomial P-convexity with applications in special function theory

Monotonicity properties and bounds involving the two-parameter generalized Grötzsch ring function

On New Modifications Governed by Quantum Hahn’s Integral Operator Pertaining to Fractional Calculus

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