Optimal bounds for Neuman-Sándor mean in terms of the convex combination of the logarithmic and the second Seiffert means

Abstract

Abstract In the article, we prove that the double inequality α L ( a , b ) + ( 1 − α ) T ( a , b ) 0 $a,b>0$ with a ≠ b $a\ne b $ if and only if α ≥ 1 / 4 $\alpha\ge1/4$ and β ≤ 1 − π / [ 4 log ( 1 + 2 ) ] $\beta\le1-\pi/[4\log(1+\sqrt{2})]$ , where NS ( a , b ) $\mathit{NS}(a,b)$ , L ( a , b ) $L(a,b)$ and T ( a , b ) $T(a,b)$ denote the Neuman-Sándor, logarithmic and second Seiffert means of two positive numbers a and b, respectively.

Keywords

• Neuman-Sándor mean
• logarithmic mean
• the second Seiffert mean