Some Inequalities of Extended Hypergeometric Functions

: Hypergeometric functions and their inequalities have found frequent applications in various ﬁelds of mathematical sciences. Motivated by the above, we set up certain inequalities including extended type Gauss hypergeometric function and conﬂuent hypergeometric function, respectively, by virtue of Hölder integral inequality and Chebyshev’s integral inequality. We also studied the monotonicity, log-concavity, and log-convexity of extended hypergeometric functions, which are derived by using the inequalities on an extended beta function.


Introduction and Preliminaries
In mathematics, the theory of special functions is an area that plays a vital role due to its applications in real analysis, functional analysis, geometry, physics, and many more subjects of science. Special functions can be defined by power series, generating functions, infinite products, and other series in orthogonal functions. In the past few years, many researchers and authors have been engaged in working on theory and applications of the special functions [1][2][3][4][5][6][7][8]. Inequalities and extensions are both important topics in the theory of special functions, but from a theoretical point of view, very few inequalities involving hypergeometric functions and extended hypergeometric functions seem to have appeared in the literature until now. Here, we aim to introduce some inequalities of extended hypergeometric functions.
Very recently, Goyal and Jain et al. [9,10] have extended the beta function, Gauss hypergeometric function, confluent hypergeometric function and studied various properties of these extended functions. They also studied the increasing or decreasing nature (monotonicity), log-concavity, and log-convexity of extended beta function in [10].
Theorem 3 ([17]). Let where, f k is not depend on b, and we consider that c > b > 0 and δ > 0, then the mapping

Proof of Theorem 4.
By the definition of the extended confluent hypergeometric function (4), we have: Now, consider a k (z) = then, we have: Now assume, Then, by the using above equation, we have: On taking x = q 1 + n, After using the Theorem (1), we have: After re-arranging the terms, we have: Then, using the above result in Equation (16), we have: i.e., increasing sequence. After the use of Lemma (1), we conclude that is increasing on (0, ∞). Hence the proof of Theorem (4), is completed.  is increasing on (0, ∞) for q 2 − q 3 ≥ 0, then by the increasing property of the function, we have: on (0, ∞).
After differentiating the function w.r.t x, we get: On some computation, we get our desired result.
Proof of Theorem 7. Consider q 1 ≥ q 1 , Then, from integral representation of the extended confluent hypergeometric function (5), we have: After using the above definition, we have: For the conditions q ≥ 0 and q 1 − q 1 ≥ 0, we can easily determine that the function f 1 (t) is decreasing and the function g 1 (t) is increasing, since function h 1 (t) is a non-negative mapping for t ∈ [0, 1].

Remark 1.
In particular, the following decreasing property of the function (35)
Proof of Theorem 8. By the similar procedure as used in the proof of Theorem (4), with some computation, we get our desired result of Theorem (8).
Theorem 9. For q 2 − q 3 ≥ 0, the extended Gauss hypergeometric function F (s) (s 1 ,s 2 ) (q 0 , q 1 , q 2 ; x) satisfied the inequality Proof of Theorem 9. On the same parallel lines as used in the proof of Theorem (5), after some computation, we get our desired result of Theorem (9). Theorem 10. For x ∈ (0, 1), the mapping is logarithmically convex on (0, ∞) Proof of Theorem 10. To prove above result, applying Theorem (3), to the extended Gauss hypergeometric function F (s) (s 1 ,s 2 ) (q 0 , q 1 , q 2 ; x). From definition of the extended Gauss hypergeometric function (2) we have: , to prove our result it sufficient to show that the On using the Theorem (2), and letting x 1 = q 1 + k, x 2 = q 1 + k − 2, y 1 = y 2 = q 2 − q 1 , we have: After re-arranging the terms, we get: Then, by using above result in Equation (39), we have: d k ≥ d k−1 , which implies (d k ) sequence is increasing. Now, we conclude by the using Theorem (3), that the mapping q 0 → F (s) (s 1 ,s 2 ) (q 0 , q 1 , q 2 ; x) is logarithmically convex on (0, ∞). Hence, proof of the Theorem (10), is completed.

Concluding Remark
We conclude our investigation by remarking that here, we describe some (presumably) new inequalities including the extended type Gauss hypergeometric function and confluent hypergeometric function, respectively. These inequalities are important for the approximation of extended confluent hypergeometric function, extended Gauss hypergeometric function, generalized Appell and Lauricella hypergeometric functions. We hope our investigation is capable of providing potential directions for future research in the approximation theory and applications of special functions.