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Inequalities of extended $(p,q)$-beta and confluent hypergeometric functions
Shahid Mubeen,Kottakkaran Sooppy Nisar,Gauhar Rahman,Muhammad Arshad 호남수학회 2019 호남수학학술지 Vol.41 No.4
In this paper, we establish the log convexity and Tur\'{a}n type inequalities of extended $(p,q)$-beta functions. Likewise, we present the log-convexity, the monotonicity and Tur\'{a}n type inequalities for extended $(p,q)$-confluent hypergeometric function by utilizing the inequalities of extended $(p,q)$-beta functions.
INEQUALITIES OF EXTENDED (p, q)-BETA AND CONFLUENT HYPERGEOMETRIC FUNCTIONS
Mubeen, Shahid,Nisar, Kottakkaran Sooppy,Rahman, Gauhar,Arshad, Muhammad The Honam Mathematical Society 2019 호남수학학술지 Vol.41 No.4
In this paper, we establish the log convexity and Turán type inequalities of extended (p, q)-beta functions. Likewise, we present the log-convexity, the monotonicity and Turán type inequalities for extended (p, q)-confluent hypergeometric function by utilizing the inequalities of extended (p, q)-beta functions.
Schwannoma of the Scrotum: Case Report and Review of the Literature
Mohammad Shahid,Syed Shamshad Ahmad,Shaista M Vasenwala,Aysha Mubeen,Sufian Zaheer,Mohammed Azfar Siddiqui 대한비뇨의학회 2014 Investigative and Clinical Urology Vol.55 No.3
Schwannomas are benign nerve sheath tumors composed of Schwann cells, which normallyproduce the insulating myelin sheath covering the peripheral nerves. Commonlocations include the head, neck, mediastinum, and retroperitoneum. These tumors areusually asymptomatic until they become large and compress the surrounding tissues. Most schwannomas occur during the third and fourth decades of life, with an equal genderdistribution. We present the case of a schwannoma that originated in the scrotum.
EXTENDED WRIGHT-BESSEL FUNCTION AND ITS PROPERTIES
Arshad, Muhammad,Mubeen, Shahid,Nisar, Kottakkaran Sooppy,Rahman, Gauhar Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.1
In this present paper, our aim is to introduce an extended Wright-Bessel function $J^{{\lambda},{\gamma},c}_{{\alpha},q}(z)$ which is established with the help of the extended beta function. Also, we investigate certain integral transforms and generalized integration formulas for the newly defined extended Wright-Bessel function $J^{{\lambda},{\gamma},c}_{{\alpha},q}(z)$ and the obtained results are expressed in terms of Fox-Wright function. Some interesting special cases involving an extended Mittag-Leffler functions are deduced.
Rahman Gauhar,Mubeen Shahid,Nisar Kottakkaran Sooppy,Choi Junesang 경남대학교 수학교육과 2019 Nonlinear Functional Analysis and Applications Vol.24 No.1
Various extensions of the Euler’s beta function have, recently, been presented and investigated. Here, choosing to use a fully extended beta function, we introduce an extended hypergeometric function, an extended confluent hypergeometric function, and an extension of the Appell function F1. We, also, use the fully extended beta function to introduce an extended Riemann-Liouville type integral operator and investigate its associated formulas and generating relations. The results presented here, being very general, can be specialized to yield some known and new results.
A NEW EXTENSION OF THE MITTAG-LEFFLER FUNCTION
Arshad, Muhammad,Choi, Junesang,Mubeen, Shahid,Nisar, Kottakkaran Sooppy,Rahman, Gauhar Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.2
Since Mittag-Leffler introduced the so-called Mittag-Leffler function in 1903, due to its usefulness and diverse applications, a variety and large number of its extensions (and generalizations) and variants have been presented and investigated. In this sequel, we aim to introduce a new extension of the Mittag-Leffler function by using a known extended beta function. Then we investigate ceratin useful properties and formulas associated with the extended Mittag-Leffler function such as integral representation, Mellin transform, recurrence relation, and derivative formulas. We also introduce an extended Riemann-Liouville fractional derivative to present a fractional derivative formula for a known extended Mittag-Leffler function, the result of which is expressed in terms of the new extended Mittag-Leffler functions.
CERTAIN FORMULAS INVOLVING A MULTI-INDEX MITTAG-LEFFLER FUNCTION
Bansal, Manish Kumar,Harjule, P.,Choi, Junesang,Mubeen, Shahid,Kumar, Devendra The Youngnam Mathematical Society 2019 East Asian mathematical journal Vol.35 No.1
Since Mittag-Leffler introduced the so-called Mittag-Leffler function, a number of its extensions have been investigated due mainly to their applications in a variety of research subjects. Shukla and Prajapati presented a lot of formulas involving a generalized Mittag-Leffler function in a systematic manner. Motivated mainly by Shukla and Prajapati's work, we aim to investigate a generalized multi-index Mittag-Leffler function and, among possible numerous formulas, choose to present several formulas involving this generalized multi-index Mittag-Leffler function such as a recurrence formula, derivative formula, three integral transformation formulas. The results presented here, being general, are pointed out to reduce to yield relatively simple formulas including known ones.
Pathway Fractional Integral Formulas Involving Extended Mittag-Leffler Functions in the Kernel
Rahman, Gauhar,Nisar, Kottakkaran Sooppy,Choi, Junesang,Mubeen, Shahid,Arshad, Muhammad Department of Mathematics 2019 Kyungpook mathematical journal Vol.59 No.1
Since the Mittag-Leffler function was introduced in 1903, a variety of extensions and generalizations with diverse applications have been presented and investigated. In this paper, we aim to introduce some presumably new and remarkably different extensions of the Mittag-Leffler function, and use these to present the pathway fractional integral formulas. We point out relevant connections of some particular cases of our main results with known results.
Certain formulas involving a multi-index Mittag-Leffler function
P. Harjule,최준상,Manish Kumar Bansal,Shahid Mubeen,Devendra Kumar 영남수학회 2019 East Asian mathematical journal Vol.35 No.1
Since Mittag-Leffler introduced the so-called Mittag-Leffler function, a number of its extensions have been investigated due mainly to their applications in a variety of research subjects. Shukla and Prajapati presented a lot of formulas involving a generalized Mittag-Leffler function in a systematic manner. Motivated mainly by Shukla and Prajapati’s work, we aim to investigate a generalized multi-index Mittag-Leffler func- tion and, among possible numerous formulas, choose to present several for- mulas involving this generalized multi-index Mittag-Leffler function such as a recurrence formula, derivative formula, three integral transformation formulas. The results presented here, being general, are pointed out to reduce to yield relatively simple formulas including known ones.
Nisar, Kottakkaran Sooppy,Rahman, Gauhar,Choi, Junesang,Mubeen, Shahid,Arshad, Muhammad The Youngnam Mathematical Society 2018 East Asian mathematical journal Vol.34 No.3
We aim to establish certain Gronwall type inequalities associated with Riemann-Liouville k- and Hadamard k-fractional derivatives. The results presented here are sure to be new and potentially useful, in particular, in analyzing dependence solutions of certain k-fractional differential equations of arbitrary real order with initial conditions. Some interesting special cases of our main results are also considered.