Convexity and inequalities related to extended beta and confluent hypergeometric functions

: In the paper, the authors establish the logarithmic convexity and some inequalities for the extended beta function and, by using these inequalities for the extended beta function, ﬁnd the logarithmic convexity and the monotonicity for the extended conﬂuent hypergeometric function.

This paper is organized as follows. In Section 2, we recall definitions of some convex functions and recite several lemmas needed in this paper. In Section 3, we present some inequalities for extended beta functions B p λ (x, y) defined in (1.3). In Section 4, we find the monotonicity and the logarithmic convexity for functions related to extended confluent hypergeometric functions Φ p (β, γ; z) defined in (1.4).

Definitions and lemmas
Now we recall definitions of some convex functions and recite several lemmas. Definition 2.1 ( [5,22]). Let X be a convex set in a real vector space and let f : X → R be a function. Then f is said to be convex on X if the inequality A function f is said to be logarithmically convex (or logarithmically concave respectively) on X if f > 0 and ln f (or − ln f respectively) is convex (or concave respectively) on X.
(2.1) 29,31]). Let θ 1 and θ 2 be positive numbers such that 1 ). Let f (x) = ∞ n=0 a n x n and g(x) = ∞ n=0 b n x n , with a n ∈ R and b n > 0 for all n, converge on (−α, α). If the sequence a n b n n≥0 is increasing (or decreasing respectively), then x → f (x) g(x) is also increasing (or decreasing respectively) on (0, α).

Inequalities for extended beta functions
Now we start off to establish inequalities for functions involving extended beta functions. . Since This can be rearranged as (3.1). The proof of Theorem 3.1 is complete.
Proof. This follows from Theorem 3.1 directly.
Proof. By the definition in (1.4), we have .
If denoting f n = a n (c) a n (d) , then .