SOME NEW INEQUALITIES FOR MEANS OF TWO ARGUMENTS

We prove certain new inequalities for special means of two arguments, including the identric, arithmetic, and geometric means. 2000 Mathematics Subject Classification. Primary 26D99, 65D32.

, if x ≠ y, I(x, x) = x, ( respectively.Let A = A(x, y) := (x + y)/2 and G = G(x, y) := √ xy denote the arithmetic and geometric means of x and y, respectively.Many interesting results are known involving inequalities between these means.For a survey of results (cf.[1,3,4,11,13,14]).Certain improvements are proved in [5,7], while connections to other means are discussed, (cf.[6,8,9,10,15]).For identities involving various means we quote the papers [6,12].In [5,8], the first author proved, among other relations, that where We note that a stronger inequality than (1.2) is (cf.[5]) but the interesting proof of (1.2), as well as the left-hand side of (1.3), is based on certain quadrature formulas (namely Simpson's and Newton's quadrature formula, respectively).As a corollary of (1.3) and (1.5), the double-inequality can be derived (see [8]).Here and throughout the rest of the paper we assume that x ≠ y.The aim of this paper is twofold.First, by applying the method of quadrature formulas, we will obtain refinements of already known inequalities (e.g., of (1.2)).Second, by using certain identities on series expansions of the considered expressions, we will obtain the best possible inequalities in certain cases (e.g., for (1.6)).

Main results
Theorem 2.1.If x and y are positive real numbers, then where r = min{x, y} and s = max{x, y}.
Remarks.Inequality (2.8) is a common generalization of (2.1) and (2.4).The lefthand side of (2.3) is a refinement of (1.2), while the left-hand side of (2.4) implies the inequality which slightly improves the right-side of (1.6).However, the best inequality of this type will be obtained by other methods.
In [6] the following identities are proved: where z = (x − y)/(x + y).Relation (2.14) is due to H.-J. Seiffert [11].With the aid of these and similar identities, strong inequalities can be deduced.We first state the following.
Theorem 2.2.The following inequalities are satisfied: (2.17) Proof.We note that (2.16) appears in [6], while the left-hand side of (2.15) has been considered in [12].We give here a unitary proof for (2.15), (2.16), and (2.17), which in fact shows that much stronger approximations may be deduced, if we want.
On the other hand, (2.12) and (2.13) yield  In [4] it is proved that log (2.24) Inequality (2.24) enabled the first author to obtain many refinements of known results (see [7]).

1 .
Introduction.The logarithmic and the identric mean of two positive real numbers x and y are defined byL = L(x, y) := y − x log y − log x , if x ≠ y, L(x, x) = x, I = I(x, y)
Therefore (2.46) cannot be true for all positive real numbers t if 0 < α < 2.