Optimal bounds for two Sándor-type means in terms of power means

Abstract

Abstract In the article, we prove that the double inequalities M α ( a , b ) 0 $a, b>0$ with a ≠ b $a\neq b$ if and only if α ≤ log 2 / [ 1 + log 2 − 2 log ( 1 + 2 ) ] = 1.5517 … $\alpha\leq\log 2/[1+\log2-\sqrt{2}\log(1+\sqrt{2})]=1.5517\ldots$ , β ≥ 5 / 3 $\beta\geq5/3$ , λ ≤ 4 log 2 / [ 4 + 2 log 2 − π ] = 1.2351 … $\lambda\leq4\log2/[4+2\log2-\pi]=1.2351\ldots$ and μ ≥ 4 / 3 $\mu\geq4/3$ , where S Q A ( a , b ) = A ( a , b ) e Q ( a , b ) / M ( a , b ) − 1 $S_{QA}(a,b)=A(a,b)e^{Q(a,b)/M(a,b)-1}$ and S A Q ( a , b ) = Q ( a , b ) e A ( a , b ) / T ( a , b ) − 1 $S_{AQ}(a,b)=Q(a,b)e^{A(a,b)/T(a,b)-1}$ are the Sándor-type means, A ( a , b ) = ( a + b ) / 2 $A(a,b)=(a+b)/2$ , Q ( a , b ) = ( a 2 + b 2 ) / 2 $Q(a,b)=\sqrt{(a^{2}+b^{2})/2}$ , T ( a , b ) = ( a − b ) / [ 2 arctan ( ( a − b ) / ( a + b ) ) ] $T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$ , and M ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / ( a + b ) ) ] $M(a,b)=(a-b)/[2\sinh ^{-1}((a-b)/(a+b))]$ are, respectively, the arithmetic, quadratic, second Seiffert, and Neuman-Sándor means.

Keywords

• Schwab-Borchardt mean
• arithmetic mean
• quadratic mean
• Neuman-Sándor mean
• second Seiffert mean
• Sándor-type mean