Abstract
G = G(x, y) = ?xy, L = L(x,y) = x?y/log(x)?log(y)'' I=I(x,y)= 1/e(xx/yy) 1/(x-y), A=A(x.y)=x+y/2, be the geometric, logarithmic, identric, and arithmetic means of x and y. We prove that the inequalities L(G2,A2) < G(L2,I2) < A(L2,I2) < I(G2,A2) are valid for all x, y > 0 with x ? y. This refines a result of Seiffert.
Highlights
Introduction and statement of the resultWe study the geometric, logarithmic, identric, and arithmetic means of two positive real numbers x and y with x = y
Logarithmic, identric, and arithmetic means of two positive real numbers x and y with x = y
Which improves the well-known inequality GA < LI given by Alzer [1]
Summary
Logarithmic, identric, and arithmetic means of two positive real numbers x and y with x = y. Many inequalities for means can be found in the literature. They obtained a double-inequality involving I(G2, A2), namely, (7) The aim of this paper is to refine the second inequality in (1.4) as follows.
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