Abstract
In this paper, we find the least value α and the greatest value β such that the double inequality $$P^{\alpha}(a,b)T^{1-\alpha}(a,b)< M(a,b)< P^{\beta}(a,b)T^{1-\beta}(a,b) $$ holds for all $a,b>0$ with $a\neq b$ , where $M(a,b)$ , $P(a,b)$ , and $T(a,b)$ are the Neuman-Sandor, the first and second Seiffert means of two positive numbers a and b, respectively.
Highlights
B > with a = b, the Neuman-Sándor mean M(a, b) [ ], the first Seiffert mean P(a, b) [ ], and the second Seiffert mean T(a, b) [ ] are defined by a–b M(a, b) = sinh– ( a–b a+b ) ( . ) P(a, b) =a√– b tan– ( a/b) – π a–b T(a, b) tan
We find the least value α and the greatest value β such that the double inequality
It can be observed that the first Seiffert mean P(a, b) can be rewritten as
Summary
The following bounds for the Seiffert means P(a, b) and T(a, b) in terms of the power mean were presented by Jagers in [ ]:. In [ ], the authors presented the following best possible Lehmer mean bounds for the Seiffert means P(a, b) and T(a, b): L– / (a, b) < P(a, b) < L (a, b) and L (a, b) < T(a, b) < L / (a, b) for all a, b > with a = b. In [ ], the authors proved that the double inequality α A(a, b) + ( – α )G(a, b) < P(a, b) < β A(a, b) + ( – β )G(a, b) holds for all a, b > with a = b if and only if α ≤ π/ , β ≥ /.
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