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http://dx.doi.org/10.4134/JKMS.j160137

MONOTONICITY PROPERTIES OF THE GENERALIZED STRUVE FUNCTIONS  

Ali, Rosihan M. (School of Mathematical Sciences Universiti Sains Malaysia)
Mondal, Saiful R. (Department of Mathematics and Statistics Collage of Science King Faisal University)
Nisar, Kottakkaran S. (Department of Mathematics College of Arts and Science Prince Sattam bin Abdulaziz University)
Publication Information
Journal of the Korean Mathematical Society / v.54, no.2, 2017 , pp. 575-598 More about this Journal
Abstract
This paper introduces and studies a generalization of the classical Struve function of order p given by $$_aS_{p,c}(x):=\sum\limits_{k=0}^{\infty}\frac{(-c)^k}{{\Gamma}(ak+p+\frac{3}{2}){\Gamma}(k+\frac{3}{2})}(\frac{x}{2})^{2k+p+1}$$. Representation formulae are derived for $_aS_{p,c}$. Further the function $_aS_{p,c}$ is shown to be a solution of an (a + 1)-order differential equation. Monotonicity and log-convexity properties for the generalized Struve function $_aS_{p,c}$ are investigated, particulary for the case c = -1. As a consequence, $Tur{\acute{a}}n$-type inequalities are established. For a = 2 and c = -1, dominant and subordinant functions are obtained for the Struve function $_2S_{p,-1}$.
Keywords
generalized Struve function; Bessel function; $Tur{\acute{a}}n$-type inequality; monotonicity properties; dominant;
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