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Register a new userInequalities of extended (p,q)-beta and confluent hypergeometric function
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In this present paper, we establish the log-convexity and Turan type inequalities of extended $(p,q)$-beta functions. Also, we present the log-convexity, the monotonicity and Turan type inequalities for extended $(p,q)$-confluent hypergeometric function by using the inequalities of extended $(p,q)$-beta functions.
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In this study our aim to define the extended $(p,q)$-Mittag-Leffler(ML) function by using extension of beta functions and to obtain the integral representation of new function. We also take the Mellin transform of this new function in terms of Wright hypergeometric function. Extended fractional derivative of the classical Mittag-Leffler(ML) function leads the extended (p,q)-Mittag-Leffler(ML) function.
Our purpose in this present paper is to investigate generalized integration formulas containing the extended generalized hypergeometric function and obtained results are expressed in terms of extended hypergeometric function. Certain special cases of the main results presented here are also pointed out for the extended Gauss hypergeometric and confluent hypergeometric functions.
Recently, various extensions and variants of Bessel functions of several kinds have been presented. Among them, the $(p,q)$-confluent hypergeometric function $Phi_{p,q}$ has been introduced and investigated. Here, we aim to introduce an extended $(p,q)$-Whittaker function by using the function $Phi_{p,q}$ and establish its various properties and associated formulas such as integral representations, some transformation formulas and differential formulas. Relevant connections of the results presented here With those involving relatively simple Whittaker functions are also pointed out.
New explicit as well as manifestly symmetric three-term summationformulas are derived for the Clausenian hypergeometric series $_3F_2(1)$ with negative integral parameter differences. Our results generalize and naturally extend several similar relations published, in recent years, by many authors. An appropriate and useful connection is established with the quite underestimated 1974 paper by P. W. Karlsson.
In this paper, we present some new inequalities for the gamma function. The main tools are the multiple-correction method developed in our previous works, and a generalized Morticis lemma.
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