Sharp Inequalities between Harmonic, Seiffert, Quadratic and Contraharmonic Means

In this paper, we present the greatest values $\alpha$, $\lambda$ and $p$, and the least values $\beta$, $\mu$ and $q$ such that the double inequalities $\alpha D(a,b)+(1-\alpha )H(a,b)<T(a,b)<\beta D(a,b)+(1-\beta )H(a,b)$, $\lambda D(a,b)+(1-\lambda)H(a,b)<C(a,b)<\mu D(a,b)+(1-\mu) H(a,b)$and$p D(a,b)+(1-p)H(a,b)<Q(a,b)<q D(a,b)+(1-q)H(a,b)$holdforall$a,b>0$with$a\neq b$,where$H(a,b)=2ab/(a+b)$,$T(a,b)=(a-b)/[2\arctan((a-b)/(a+b))]$,$Q(a,b)=\sqrt{(a^2+b^2)/2}$,$C(a,b)=(a^2+b^2)/(a+b)$and$D(a,b)=(a^3+b^3)/(a^2+b^2)$ are the harmonic, Seiffert, quadratic, first contraharmonic and second contraharmonic means of $a$ and $b$, respectively.