Some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function

In the paper, the authors present some inequalities involving the extended gamma function and the Kummer confluent hypergeometric k-function via some classical inequalities such as Chebychev’s inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the log-convexity of the extended gamma function by using the Hölder inequality, and introduce a Turán type mean inequality for the Kummer confluent k-hypergeometric function.

The beta function B(x, y) can be defined [18,21,22]  In 1995, Chaudhry and Zubair [4] introduced the extended gamma function, If b = 0, then b becomes the classical gamma function .
In 1997, Chaudhry et al. [3] introduced the extended beta function, It is clear that B 0 (x, y) = B(x, y).
In 2009, Barnard et al. [1] established three inequalities where A(α, β) = α+β 2 and G(α, β) = √ αβ are the arithmetic and geometric means and is the Kummer confluent hypergeometric function [25,28]. The Kummer confluent hypergeometric k-function is defined by where (a) n,k = a(a + k)(a + 2k) · · · a + (n -1)k for n ≥ 1 and k > 0 with (a) 0,k = 1 is the Pochhammer k-symbol, which can also be rewritten as (a) n,k = k (a + nk) k (a) and the gamma k-function k (a) is defined [6] by In 2012, Mubeen [15] introduced the k-analogue of Kummer's transformation as In Sect. 2, we prepare two lemmas. In Sect. 3, we discuss applications of some integral inequalities such as Chebychev's integral inequality. In Sect. 4, we prove the logarithmic convexity of the extended gamma function. In the last section, we introduce a mean inequality of Turán type for the Kummer confluent hypergeometric k-function.

Lemmas
In order to obtain our main results, we need the following lemmas. [7,8,12,23]

Lemma 2.1 (Chebychev's integral inequality
, and h(x)g(x) are integrable on I. If f (x) and g(x) are synchronous (or asynchronous, respectively) on I, that is, for all x, y ∈ I, then Lemma 2.2 (Hölder's inequality [29,30]) Let p and q be positive real numbers such that

Inequalities involving the extended gamma function via Chebychev's integral inequality
In this section, we prove some inequalities involving the extended gamma function via Chebychev's integral inequality in Lemma 2.1. If r(pmr) 0, then we can claim that the mappings f and g are synchronous (asynchronous) on (0, ∞). Thus, by applying Chebychev's inequality on I = (0, ∞) to the functions f , g and h defined above, we can write This implies that By (1.1), we acquire the required inequality (3.1).
Proof By setting m = p and r = q in Theorem 3.1, we obtain r(pmr) = -q 2 ≤ 0 and then the inequality (3.1) provides the desired Corollary 3.1.
Now if the condition (m -1)(n -1) 0 holds, then Chebychev's integral inequality applied to the functions f , g, and h means This implies that By the definition of the extended gamma function, we have .
The required proof is complete.
If the conditions of Theorem 3.1 hold, then the mappings f and g are synchronous (asynchronous) on [0, ∞). Thus, by applying Chebychev's integral inequality in Lemma 2.1 to the functions f , g and h defined above, we have This implies that Thus by the definition of extended gamma function, we have The required proof is complete.

Log-convexity of the extended gamma function
It is well known that, if f > 0 and ln f is convex, then f is said to be a logarithmically convex function. Every logarithmically convex must be convex. See [16] and [19,Remark 1.9]. In this section, we verify the log-convexity of extended gamma function. Proof Let p and q be positive numbers such that 1 As a result, the function b is logarithmically convex.

A mean inequality of the Turán type for the Kummer confluent hypergeometric k-function
In this section, we present a mean inequality involving the confluent hypergeometric kfunction. For this purpose, we consider the relation is valid for all nonzero x ∈ R.
First proof Assume that x > 0. For c = 0, -1, -2, . . . , define From (5.1), it follows that Accordingly, by the Cauchy product, we have If s is even, then Accordingly, Carefully simplifying gives where ψ k = k k is the digamma k-function (see [6,11,16]). Hence, the function h k is increasing under the condition stated. This fact together with the aid of (5.3) and (5.4) yields (T s,s-r,k -T s,r,k )(s -2r)x s > 0, (5.5) where a ≥ v ≥ 0, x > 0, c + k > 0, and c = 0. Consequently, from (5.5), it follows that Therefore, the function f v,k is absolutely monotonic on (0, ∞), that is, f ( ) v,k (x) > 0 for = 0, 1, 2, . . . . This proves Theorem 5.1 for the case x > 0. Now suppose that x < 0, a, b > 0, and v ∈ N with a, b ≥ vk. Since φ k (a, c, x) is symmetric in a and b, by interchanging a and b in Theorem 5.1, we obtain By using Kummer's transformation (1.2), we have Thus, Theorem 5.1 also holds for x < 0.
Second proof Since (a) n,k = a(a + k)(a + 2k) · · · a + (n -1)k = k n a k a k + 1 a k + 2 · · · a k + (n -1) = k n a k n , it follows that Replacing a and b by a k and b k , respectively, gives Theorem 5.1.

Corollary 5.1
If a > 0 and c + k > 0 with c = 0, then the inequality Proof This follows directly from the proof of Theorem 5.1 and the fact that Eq. (5.6) holds under the conditions c + k > 0 and c = 0.

Corollary 5.2 If v ∈ N and a, b ≥ v, then
for all nonzero x ∈ R, where A and G are, respectively, the arithmetic and geometric means.  a + b, x).
For x ≥ 0, the right hand side inequality in (5.7) follows from taking square root of (5.2). The proof of Corollary 5.2 for x ≥ 0 is thus complete.
Now assume x < 0 with a, b ≥ v. Interchanging a and b in (5.7) one arrives at Making use of the k-analogue of Kummer's transformation and the homogeneity of A and G acquires Consequently, Theorem (5.7) also follows for x < 0.
Remark 5.1 In Sect. 5, we have established a Turán type and mean inequality for kanalogue of the Kummer confluent hypergeometric function. If we let k → 1, then we can conclude to the corresponding inequalities of the confluent hypergeometric function.
Remark 5.2 In [2], some inequalities of the Turán type for confluent hypergeometric functions of the second kind were also discovered.
Remark 5.3 By the way, we note that Refs. [9,10,13,14,26,32,33] belong to the same series in which inequalities and complete monotonicity for functions involving the gamma function (x) and the logarithmic function ln(1 + x) were discussed.
Remark 5.4 This paper is a slightly revised version of the preprint [17].

Conclusions
In this paper, we present some inequalities involving the extended gamma function b (z) via some classical inequalities such as Chebychev's inequality for synchronous (or asynchronous, respectively) mappings, give a new proof of the log-convexity of the extended gamma function b (z) by using the Hölder inequality, and introduce a Turán type mean inequality for the Kummer confluent k-hypergeometric function φ(z).