Optimal Lehmer Mean Bounds for the Combinations of Identric and Logarithmic Means

For any , we answer the questions: what are the greatest values and and the least values and , such that the inequalities and hold for all with ? Here, , , and denote the identric, logarithmic, and th Lehmer means of two positive numbers and , respectively. 1. Introduction For with , the logarithmic mean and identric mean are defined by respectively. In the recent past, both mean values have been the subject of intensive research. In particular, many remarkable inequalities for and can be found in the literature [1–19]. In [14, 17, 20], inequalities between , , and the classical arithmetic-geometric mean of Gauss are proved. The ratio of identric means leads to the weighted geometric mean which has been investigated in [11, 13, 21]. It might be surprising that the logarithmic mean has applications in physics, economics, and even in meteorology [22–24]. In [22], the authors study a variant of Jensen’s functional equation involving , which appears in a heat conduction problem. A representation of as an infinite product and an iterative algorithm for computing the logarithmic mean as the common limit of two sequences of special geometric and arithmetic means are given in [5]. In [25, 26], it is shown that can be expressed in terms of Gauss’ hypergeometric function . And in [26], the authors prove that the reciprocal of the logarithmic mean is strictly totally positive; that is, every determinant with elements , where and , is positive for all . For , the power mean and Lehmer mean of order of two positive numbers and with are defined by It is well known that and are strictly increasing with respect to for fixed with . The main properties for and are given in [27–32]. Let , , and be the arithmetic, geometric, and harmonic means of two positive numbers and , respectively. Then it is well known that for all with . The following sharp bounds for , , , and in terms of power means are proved in [2–4, 6, 8, 9, 33]: for all with . Alzer and Qiu [19] proved that for all with if and only if , , and . The following sharp upper and lower Lehmer mean bounds for , , , and are presented in [34]: for all with . The purpose of this paper is to present the best possible upper and lower Lehmer mean bounds of the product and the sum for any and all with . 2. Lemmas In order to establish our main result, we need several lemmas, which we present in this section. Lemma 1. If , then the following statements are true:(1) ;(2) ;(3) ;(4) ;(5) ;(6) . Proof. (1) We clearly see that (2) Let Then simple computations lead to Therefore, Lemma 1(2) follows from (14) and (15). (3) Let Then