Inequalities for the generalized weighted mean values of g -convex functions with applications
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Inequalities for the generalized weighted mean values of g-convex functions with applications Ming-Bao Sun1 · Yu-Ming Chu2,3 Received: 26 February 2020 / Accepted: 13 July 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In the article, we establish several inequalities for the generalized weighted mean values of g-convex functions. As applications, we provide several new Hermite-Hadamard type inequalities for the g-convex functions. Our results are the generalizations of some previously known results. Keywords r -convex function · g-convex function · Generalized weighted mean values · Hermite-Hadamard type inequality Mathematics Subject Classification 26D15 · 26A51
1 Introduction Let I ⊆ R be an interval. Then a real-valued function f : I → R is said to be convex (concave) on I if the inequality f (t x + (1 − t)y) ≤ (≥)t f (x) + (1 − t) f (y) holds for all x, y ∈ I and t ∈ [0, 1]. It is well known that the convex functions have wide applications in pure and applied mathematics [1–20]. In particular, many remarkable inequalities for the convex functions can be found in the literature [21–42]. Recently, a great deal of generalizations, extensions and variants have been made for the convexity, for example, p-convexity [43], H p,q convexity [44], ρ-convexity [45], generalized convexity [46], h-convexity [47], GG- and G A-convexities [48], exponential convexity [49], s-convexity [50,51] and so on.
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Yu-Ming Chu [email protected] Ming-Bao Sun [email protected]
1
School of Mathematics, Hunan Institute of Science and Technology, Yueyang 414000, China
2
Department of Mathematics, Huzhou University, Huzhou 313000, China
3
Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha University of Science and Technology, Changsha 410114, China 0123456789().: V,-vol
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M. Sun, Y. Chu
Let a, b ∈ R with a < b and f : [a, b] → R be a convex function. Then the classical Hermite-Hadamard inequality [52–57] states that the double inequality b a+b f (a) + f (b) 1 f (t)dt ≤ f ≤ (1.1) 2 b−a a 2 holds. In the past 20 years, the Hermite–Hadamard inequality (1.1) has been generalized, refined and improved by many researchers [58–61]. Now, we recall some basic definitions which are needed in the article. Definition 1.1 Let I ⊆ R be an interval, r ∈ R and f : I → (0, ∞) be a positive real-valued function. Then f is said to be r -convex (concave) on I if the inequality 1/r λ f r (x) + (1 − λ) f r (y) , r = 0, f [λx + (1 − λ)y] ≤ (≥) (1.2) f λ (x) f 1−λ (y), r =0 holds for all x, y ∈ I and λ ∈ [0, 1]. From (1.2) we clearly see that 0-convex function is the log-convex function and 1-convex function is the ordinary convex function. Definition 1.2 Let a, b ∈ R with a < b, f : [a, b] → R be a real-valued function, J = { f (x)|x ∈ [a, b]} be the range of f on [a, b], and g be a continuous and strictly monotone function defined on J . Then f is said to be g-convex (concave) if the inequality f [λx + (1 − λ)y] ≤ (≥)g −1 [λg( f (x)) + (1 − λ)g( f (y))] is v
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