Abstract
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.
Highlights
We find the logarithmic convexity and the monotonicity related to the extended confluent hypergeometric function Φλp(β, γ; z) defined in (1.4)
The decreasing property of the function β is equivalent to the inequality
This paper is a slightly revised version of the preprint [26]
Summary
In [30], the extended beta function B(x, y; p) defined by (1.1) was generalized as. B1p(x, y) = B(x, y; p) and B11(x, y) = B(x, y) It is well known [11, 15] that, when λ ∈ [0, 1], the Mittag-Leffler function Eλ(−w) is completely monotonic on (0, ∞). In [30], the extended confluent hypergeometric function Φp(β, γ; z) defined by (1.2) was generalized as.
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