Sharp inequalities involving Neuman-Sándor and logarithmic means (Q2855179)

Let \(a,b>0,\) \(a\neq b\) and \(v=(a-b)/(a+b).\) In this paper, the following bivariate means are discussed: the harmonic mean \(H,\) geometric mean \(G,\) arithmetic mean \(A,\) root-square mean \(Q,\) contra-harmonic mean \(C,\) logarithmic mean \(L\) and the Neuman-Sándor mean \(M\) defined as functions of \(a\) and \(b\) by NEWLINE\[NEWLINEH=\frac{2ab}{a+b},G=\sqrt{ab},A=\frac{a+b}{2},Q=\sqrt{\frac{a^{2}+b^{2}}{2}} ,C=\frac{a^{2}+b^{2}}{a+b},NEWLINE\]NEWLINE NEWLINE\[NEWLINEL=A\frac{v}{\tanh ^{-1}v}\text{ and }M=A\frac{v}{\sinh ^{-1}v},NEWLINE\]NEWLINE \medskipNEWLINE NEWLINErespectively. All these means are comparable and NEWLINE\[NEWLINEH<G<L<A<M<Q<C.NEWLINE\]NEWLINE For any mean \(N\) and any parameter \(p,\) \(|p|\leq 1,\) the author defines a mean \(N_{p}\) as follows: \(N_{p}(a,b)=N(x,y)\) where \( x=wa+(1-w)b,\) \(y=wb+(1-w)a,\) \(w=(1+p)/2\). The main results of the paper are two-sided sharp bounds for the means \(M\) and \(L\) via \( Q_{p},\) \(C_{p},\) \(H_{p}\) or \(G_{p}.\) For example the two-sided inequality NEWLINE\[NEWLINEH_{p}<L<H_{q}NEWLINE\]NEWLINE holds true if and only if \(p=1\) and \(q\leq 1/\sqrt{3}.\)