Optimal convex combination bounds of geometric and Neuman means for Toader-type mean

Abstract

Abstract In this paper, we prove that the double inequalities α N Q A ( a , b ) + ( 1 − α ) G ( a , b ) 0 $a,b>0$ with a ≠ b $a\neq b$ if and only if α ≤ 3 / 8 $\alpha \leq 3/8$ , β ≥ 4 / [ π ( log ( 1 + 2 ) + 2 ) ] = 0.5546 ⋯ $\beta \geq 4/ [\pi ( \log (1+\sqrt{2})+\sqrt{2}) ]=0.5546 \cdots $ , λ ≤ 3 / 10 $\lambda \leq 3/10$ and μ ≥ 8 / [ π ( π + 2 ) ] = 0.4952 ⋯ $\mu \geq 8/ [\pi (\pi +2) ]=0.4952 \cdots $ , where T D ( a , b ) $TD(a,b)$ , G ( a , b ) $G(a,b)$ , A ( a , b ) $A(a,b)$ and N Q A ( a , b ) $N_{QA}(a,b)$ , N A Q ( a , b ) $N_{AQ}(a,b)$ are the Toader, geometric, arithmetic and two Neuman means of a and b, respectively.

Keywords

• Toader mean
• geometric mean
• Neuman mean