The Monotonicity Results for the Ratio of Certain Mixed Means and Their Applications

We continue to adopt notations and methods used in the papers illustrated by Yang (2009, 2010) to investigate the monotonicity properties of the ratio of mixed two-parameter homogeneous means. As consequences of our results, the monotonicity properties of four ratios of mixed Stolarsky means are presented, which generalize certain known results, and some known and new inequalities of ratios of means are established. 1. Introduction Since the Ky Fan [1] inequality was presented, inequalities of ratio of means have attracted attentions of many scholars. Some known results can be found in [2–14]. Research for the properties of ratio of bivariate means was also a hotspot at one time. In this paper, we continue to adopt notations and methods used in the paper [13, 14] to investigate the monotonicity properties of the functions defined by where the , with , is the so-called two-parameter homogeneous functions defined by [15, 16]. For conveniences, we record it as follows. Definition 1.1. Let : be a first-order homogeneous continuous function which has first partial derivatives. Then, is called a homogeneous function generated by with parameters and if is defined by for where and denote first-order partial derivatives with respect to first and second component of , respectively. If exits and is positive for all , then further define and . Remark 1.2. Witkowski [17] proved that if the function is a symmetric and first-order homogeneous function, then for all is a mean of positive numbers and if and only if is increasing in both variables on . In fact, it is easy to see that the condition “ is symmetric” can be removed. If is a mean of positive numbers and , then it is called two-parameter homogeneous mean generated by . For simpleness, is also denoted by or . The two-parameter homogeneous function generated by is very important because it can generates many well-known means. For example, substituting if with and for yields Stolarsky means defined by where if , with , and is the identric (exponential) mean (see [18]). Substituting for yields Gini means defined by where (see [19]). As consequences of our results, the monotonicity properties of four ratios of mixed Stolarsky means are presented, which generalize certain known results, and some known and new inequalities of ratios of means are established. 2. Main Results and Proofs In [15, 16, 20], two decision functions play an important role, that are, In [14], it is important to another key decision function defined by Note that the function defined by has well properties (see [15, 16]). And it has shown in