Abstract
Some new inequalities of Hermite-Hadamard type for GA-convex functions defined on positive intervals are given. Refinements and weighted version of known inequalities are provided. Some applications for special means are also obtained.
Highlights
Let J ⊂ (0, ∞) be an interval; a real-valued function f : J → R is said to be GA-convex on J, if (1.1)f x1−λyλ ≤ (≥) (1 − λ) f (x) + λf (y) for all x, y ∈ J and λ ∈ [0, 1]
It is known that the function f (x) = ln (1 + x) is GA-convex on (0, ∞) [2]
It has been shown in [28] that the function f : (0, ∞) → R defined by 1 f (x) = ψ (x) +
Summary
Let J ⊂ (0, ∞) be an interval; a real-valued function f : J → R is said to be GA-convex (concave) on J (see for instance [2]), if (1.1). For real and positive values of x, the Euler gamma function Γ and its logarithmic derivative ψ, the so-called digamma function, are defined by. It has been shown in [28] that the function f : (0, ∞) → R defined by 1 f (x) = ψ (x) +. 2x is GA-concave on (0, ∞) while the function g : (0, ∞) → R defined by 11 g (x) = ψ (x) + 2x + 12x2 is GA-convex on (0, ∞). If b > a > 0 and f : [a, b] → R is a differentiable GA-convex (concave) function on [a, b] , (1.4). The differentiability of the function is not necessary in Theorem 1 for the first inequality (1.4) to hold, as shown in [10]
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