scholarly journals Functional Inequalities and Monotonicity Results for Modified Lommel Functions of the First Kind

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Robert E. Gaunt
Keyword(s):  
Functional Inequalities ◽  
Lommel Functions ◽  
Modified Struve Function ◽  
Turán Type Inequalities ◽  
Monotonicity Results

AbstractWe establish some monotonicity results and functional inequalities for modified Lommel functions of the first kind. In particular, we obtain new Turán type inequalities and bounds for ratios of modified Lommel functions of the first kind, as well as the function itself. These results complement and in some cases improve on existing results, and also generalise a number of the results from the literature on monotonicity patterns and functional inequalities for the modified Struve function of the first kind.

scholarly journals Inequalities for Some Integrals Involving Modified Lommel Functions of the First Kind

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Robert E. Gaunt
Keyword(s):  
Hypergeometric Function ◽  
Generalized Hypergeometric Function ◽  
Double Inequality ◽  
Lommel Functions ◽  
Modified Struve Function ◽  
Lommel Function

AbstractIn this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $$t_{\mu ,\nu }(x)$$tμ,ν(x). In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function $$t_{\mu ,\nu }(x)$$tμ,ν(x), for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.

scholarly journals On generalized Bessel–Maitland function

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Hanaa M. Zayed
Keyword(s):  
Beta Function ◽  
Main Idea ◽  
Forthcoming Paper ◽  
Residue Theorem ◽  
Maclaurin Series ◽  
Hyperbolic Functions ◽  
Turán Type Inequalities ◽  
Recurrence Formulae ◽  
Different Order ◽  
Monotonicity Results

AbstractAn approach to the generalized Bessel–Maitland function is proposed in the present paper. It is denoted by $\mathcal{J}_{\nu , \lambda }^{\mu }$ J ν , λ μ , where $\mu >0$ μ > 0 and $\lambda ,\nu \in \mathbb{C\ }$ λ , ν ∈ C get increasing interest from both theoretical mathematicians and applied scientists. The main objective is to establish the integral representation of $\mathcal{J}_{\nu ,\lambda }^{\mu }$ J ν , λ μ by applying Gauss’s multiplication theorem and the representation for the beta function as well as Mellin–Barnes representation using the residue theorem. Moreover, the mth derivative of $\mathcal{J}_{\nu ,\lambda }^{\mu }$ J ν , λ μ is considered, and it turns out that it is expressed as the Fox–Wright function. In addition, the recurrence formulae and other identities involving the derivatives are derived. Finally, the monotonicity of the ratio between two modified Bessel–Maitland functions $\mathcal{I}_{\nu ,\lambda }^{\mu }$ I ν , λ μ defined by $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)=i^{-2\lambda -\nu }\mathcal{J}_{ \nu ,\lambda }^{\mu }(iz)$ I ν , λ μ ( z ) = i − 2 λ − ν J ν , λ μ ( i z ) of a different order, the ratio between modified Bessel–Maitland and hyperbolic functions, and some monotonicity results for $\mathcal{I}_{\nu ,\lambda }^{\mu }(z)$ I ν , λ μ ( z ) are obtained where the main idea of the proofs comes from the monotonicity of the quotient of two Maclaurin series. As an application, some inequalities (like Turán-type inequalities and their reverse) are proved. Further investigations on this function are underway and will be reported in a forthcoming paper.

scholarly journals Turán-Type Inequalities for Some Lommel Functions of the First Kind

2015 ◽  
Vol 59 (3) ◽  
pp. 569-579 ◽  
Author(s):  
Árpád Baricz ◽  
Stamatis Koumandos
Keyword(s):  
Classical Result ◽  
Entire Functions ◽  
Infinite Product ◽  
Product Representation ◽  
Lommel Functions ◽  
Turán Type Inequalities ◽  
Infinite Product Representation

AbstractIn this paper certain Turán-type inequalities for some Lommel functions of the first kind are deduced. The key tools in our proofs are the infinite product representation for these Lommel functions of the first kind, a classical result of Pólya on the zeros of some particular entire functions, and the connection of these Lommel functions with the so-called Laguerre–Pólya class of entire functions. Moreover, it is shown that in some cases Steinig's results on the sign of Lommel functions of the first kind combined with the so-called monotone form of l’Hospital's rule can be used in the proof of the corresponding Turán-type inequalities.

scholarly journals On Riemann—Liouville and Caputo Fractional Forward Difference Monotonicity Analysis

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1303
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Faraidun Kadir Hamasalh
Keyword(s):  
Time Scale ◽  
Initial Value Problems ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Discrete Fractional Calculus ◽  
Initial Value ◽  
Forward Difference ◽  
The Mean ◽  
Monotonicity Analysis ◽  
Monotonicity Results

Monotonicity analysis of delta fractional sums and differences of order υ∈(0,1] on the time scale hZ are presented in this study. For this analysis, two models of discrete fractional calculus, Riemann–Liouville and Caputo, are considered. There is a relationship between the delta Riemann–Liouville fractional h-difference and delta Caputo fractional h-differences, which we find in this study. Therefore, after we solve one, we can apply the same method to the other one due to their correlation. We show that y(z) is υ-increasing on Ma+υh,h, where the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and then, we can show that y(z) is υ-increasing on Ma+υh,h, where the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) for each z∈Ma+h,h. Conversely, if y(a+υh) is greater or equal to zero and y(z) is increasing on Ma+υh,h, we show that the delta Riemann–Liouville fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to zero, and consequently, we can show that the delta Caputo fractional h-difference of order υ of a function y(z) starting at a+υh is greater or equal to −1Γ(1−υ)(z−(a+υh))h(−υ)y(a+υh) on Ma,h. Furthermore, we consider some related results for strictly increasing, decreasing, and strictly decreasing cases. Finally, the fractional forward difference initial value problems and their solutions are investigated to test the mean value theorem on the time scale hZ utilizing the monotonicity results.

scholarly journals Functional Inequalities for Two-Level Concentration

2021 ◽  
Author(s):  
Franck Barthe ◽  
Michał Strzelecki
Keyword(s):  
Sobolev Inequality ◽  
Poincaré Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Probability Measures ◽  
Exponential Rate ◽  
Functional Inequalities ◽  
Concentration Inequality ◽  
Poincare Inequality ◽  
Free Concentration ◽  
Concentration Property

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

On a certain stability of the Hermite–Hadamard inequality

2008 ◽  
Vol 465 (2102) ◽  
pp. 571-583 ◽  
Author(s):  
Attila Házy ◽  
Zsolt Páles
Keyword(s):  
Functional Inequalities ◽  
Hadamard Inequality ◽  
Real Functions ◽  
The Stability

The classical Hermite–Hadamard inequality, under some regularity assumptions, characterizes convexity of real functions. The aim of this paper is to establish connections between the stability forms of the functional inequalities related to Jensen convexity, convexity and the Hermite–Hadamard inequality.

scholarly journals L q -Functional Inequalities and Weighted Porous Media Equations

2007 ◽  
Vol 28 (1) ◽  
pp. 35-59 ◽  
Author(s):  
Jean Dolbeault ◽  
Ivan Gentil ◽  
Arnaud Guillin ◽  
Feng-Yu Wang
Keyword(s):  
Porous Media ◽  
Functional Inequalities ◽  
Porous Media Equations
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