Hermite-Hadamard and Simpson Type Inequalities for Differentiable P-GA-Functions

The author introduces the concept of the -GA-functions, gives Hermite-Hadamard's inequalities for -GA-functions, and defines a new identity. By using this identity, the author obtains new estimates on generalization of Hadamard and Simpson type inequalities for -GA-functions. Some applications to special means of real numbers are also given. 1. Introduction Let real function be defined on some nonempty interval of real line . The function is said to be convex on if inequality holds for all and . We recall that a function is said to be -function on or belong to the class if it is nonnegative and for all and . Note that contain all nonnegative convex and quasiconvex functions [1]. The following inequalities are well known in the literature as Hermite-Hadamard inequality and Simpson inequality, respectively. Theorem 1. Let be a convex function defined on the interval of real numbers and with . The following double inequality holds: Theorem 2. Let be a four times continuously differentiable mapping on and . Then the following inequality holds: Definition 3 (see [2, 3]). A function is said to be GA-convex (geometric-arithmetically convex) if for all and . In recent years, many authors have studied errors estimations for Hermite-Hadamard and Simpson inequalities; for refinements, counterparts, and generalization concerning -functions and GA-convex, see [4–11]. In this paper, the concept of the -GA-function is introduced, Hermite-Hadamard’s inequalities for -GA-functions are established, and a new identity for differentiable functions is defined. By using this identity, the author obtains a generalization of Hadamard and Simpson type inequalities for -GA-functions. 2. Main Results Let be a differentiable function on , the interior of ; throughout this section we will take where with and . Definition 4. A function is said to be -GA-function ( -geometric-arithmetic function) on if for any and . Proposition 5. Let . If is -function and nondecreasing, then is -GA-function on . Proof. This follows from for all and . Proposition 6. Let . If is -GA-function and nonincreasing, then is -function on . Proof. The conclusion follows from for all and , respectively. Hermite-Hadamard’s inequalities can be represented for -GA-functions as follows. Theorem 7. Let be a function such that ( is integrable on ), where with . If is a -GA-function on , then the following inequalities hold: with . Proof. Since is a -GA-function on , we have for all (with in inequality (7)) Choosing ,？？ , we get Integrating the resulting inequality with respect to over , we obtain and the first