engleski

Neuman-Sandor mean, asymptotic expansions and related inequalities

The subject of this paper is a systematic study of inequalities of the form $$ (1-\mu) M_1+\mu M_3 \leq M_2 \leq (1-\nu) M_1+\nu M_3 $$ which cover Neuman-S\'andor mean and some classical means. Furthermore, following inequalities with optimal parameters were proved: $$ \mu\frac1{; ; ; ; H(s, t)}; ; ; ; +(1-\mu)\frac1{; ; ; ; N\!S(s, t)}; ; ; ; \leq \frac1{; ; ; ; A(s, t)}; ; ; ; \leq \nu\frac1{; ; ; ; H(s, t)}; ; ; ; +(1-\nu)\frac1{; ; ; ; N\!S(s, t)}; ; ; ; $$ and $$ \mu\frac1{; ; ; ; H(s, t)}; ; ; ; +(1-\mu)\frac1{; ; ; ; N(s, t)}; ; ; ; \leq \frac1{; ; ; ; N\!S(s, t)}; ; ; ; \leq \nu\frac1{; ; ; ; H(s, t)}; ; ; ; +(1-\nu)\frac1{; ; ; ; N(s, t)}; ; ; ; . $$

Neuman-Sandor mean; asymptotic expansion; optimal convex combination

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