**Abstract.** In this paper we study inequalities of the form

(1 - μ) *M*_{1}(*s*, *t*) + μ *M*_{3}(*s*, *t*)
≤ *M*_{2}(*s*, *t*)
≤ (1 - ν) *M*_{1}(*s*, *t*) + ν *M*_{3}(*s*, *t*),

**2010 Mathematics Subject Classification.**
26E60, 41A60.

**Key words and phrases.** Seiffert means, asymptotic expansion, optimal convex
combination.

**References:**

- M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions,
National Bureau of Standards, Washington, DC, 1964.

MathSciNet - J. Aczél and Z. Páles,
*The behaviour of means under equal increments of their variables*, Amer. Math. Monthly**95**(1988), 856-860.

MathSciNet CrossRef - P. S. Bullen,
*Averages still on the move*, Math. Mag.**63**(1990), 250-255.

MathSciNet CrossRef - P. S. Bullen,
Handbook of Means and Their Inequalities,
Kluwer Academic Publishers, Dordrecht, 2003.

MathSciNet CrossRef - P. S. Bullen, D. S. Mitrinović and P. M. Vasić,
Means and their inequalities,
D. Reidel Publishing Co., Dordrecht, 1988.

MathSciNet CrossRef - C.-P. Chen, N. Elezović and L. Vukšić,
*Asymptotic formulae associated with the Wallis power function and digamma function*, J. Class. Anal.**2**(2013), 151-166.

MathSciNet - Y. M. Chu, Y. F. Qiu, M. K. Wang and G. D. Wang,
*The Optimal convex combination bounds of arithmetic and harmonic means for the Seiffert's mean*, J. Inequal. Appl.**2010**, Art. ID 436457, 7 pp.

MathSciNet CrossRef - Y. M. Chu, M. K. Wang and W. M. Gong,
*Two sharp double inequalities for Seiffert mean*, J. Inequal. Appl.**2011**, 2011:44, 7 pp.

MathSciNet CrossRef - Y. M. Chu, C. Zong and G. D. Wang,
*Optimal convex combination bounds of Seiffert and geometric means for the arithmetic mean*, J. Math. Inequal.**5**(2011), 429-434.

MathSciNet CrossRef - N. Elezović,
*Asymptotic inequalities and comparison of classical means*, J. Math. Inequal.**9**(2015), 177-196.

MathSciNet CrossRef - N. Elezović and L. Vukšić,
*Asymptotic expansions of bivariate classical means and related inequalities*, J. Math. Inequal.**8**(2014), 707-724.

MathSciNet CrossRef - A. Erdélyi,
Asymptotic expansions, Dover Publications, 1956.

MathSciNet - S.-Q. Gao, H.-Y. Gao and W.-Y. Shi,
*Optimal convex combination bounds of the centroidal and harmonic means for the Seiffert mean*, Int. J. Pure Appl. Math.**70**(2011), 701-709. - L. Hoehn and I. Niven,
*Averages on the move*, Math. Mag.**58**(1985), 151-156.

MathSciNet CrossRef - W.-D. Jiang and J. Cao and F. Qi,
*Two sharp inequalities for bounding the Seiffert mean by the arithmetic, centroidal, and contra-harmonic means*,

`arXiv:1201.6432v1 [math.CA]` - W.-D. Jiang and F. Qi,
*Some sharp inequalities involving Seiffert and other means and their concise proofs*, Math. Inequal. Appl.**15**(2012), 1007-1017.

MathSciNet CrossRef - H. Liu and X. J. Meng,
*The optimal convex combination bounds for Seiffert's mean*, J. Inequal. Appl.**2011**, Art. ID 686834, 9 pp.

MathSciNet CrossRef - E. Neuman,
*A one-parameter family of bivariate means*, J. Math. Inequal.**7**(2013), 399-412.

MathSciNet CrossRef