Seiffert Mean Research Articles

Several Double Inequalities for Integer Powers of the Sinc and Sinhc Functions with Applications to the Neuman–Sándor Mean and the First Seiffert Mean

In the paper, the authors establish a general inequality for the hyperbolic functions, extend the newly-established inequality to trigonometric functions, obtain some new inequalities involving the inverse sine and inverse hyperbolic sine functions, and apply these inequalities to the Neuman–Sándor mean and the first Seiffert mean.

Sharp generalized Seiffert mean bounds for the Toader mean of order 4

In the article, we present the best possible parameters $$\alpha ,\beta $$ such that the double inequality $$\begin{aligned} S_{\alpha }(a,b)<T_{4}(a,b)<S_{\beta }(a,b) \end{aligned}$$ holds for $$a, b>0$$ with $$a\ne b$$ , and provide new bounds for the complete elliptic integral of the second kind, where $$S_{p}(a,b)$$ and $$T_{4}(a,b)$$ are the generalized Seiffert mean and Toader mean of order 4, respectively.

Several sharp inequalities about the first Seiffert mean

In this paper, we deal with the problem of finding the best possible bounds for the first Seiffert mean in terms of the geometric combination of logarithmic and the Neuman–Sándor means, and in terms of the geometric combination of logarithmic and the second Seiffert means.

Optimal bounds for the first Seiffert mean in terms of the convex combination of the logarithmic and Neuman-Sándor mean

Optimal bounds for the first Seiffert mean in terms of the convex combination of the logarithmic and Neuman-Sándor mean

Sub-super-stabilizability of certain bivariate means via mean-convexity

In this paper, we first show that the first Seiffert mean P is concave whereas the second Seiffert mean T and the Neuman-Sandor mean NS are convex. As applications, we establish the sub-stabilizability/super-stabilizability of certain bivariate means. Open problems are derived as well.

Index of a bivariate mean and applications

Exploring some results of (Raissouli in J. Math. Inequal. 10(1):83-99, 2016) from another point of view, we introduce here some power-operations for (bivariate) means. As application, we construct some classes of means in one or two parameters including some standard means. We also define a law between means which allows us to obtain, among others, a simple relationship involving the three familiar means, namely the first Seiffert mean, the second Seiffert mean, and the Neuman-Sandor mean. At the end, more examples of interest are discussed and open problems are derived as well.

Sharp inequalities for bounding Seiffert mean in terms of the arithmetic, centroidal, and contra-harmonic means

AbstractIn the paper, the authors find two sharp and double inequalities for bounding the second Seiffert mean either by a one-parameter family of means derived from the centroidal mean or by a convex combination of the arithmetic and contra-harmonic means.

The first Seiffert mean is strictly ( G , A ) -super-stabilizable

The concept of strictly super-stabilizability for bivariate means has been defined recently by Raisoulli and Sandor (J. Inequal. Appl. 2014:28, 2014). We answer into affirmative to an open question posed in that paper, namely: Prove or disprove that the first Seiffert mean P is strictly -super-stabilizable. We use series expansions of the functions involved and reduce the main inequality to three auxiliary ones. The computations are performed with the aid of the computer algebra systems Maple and Maxima. The method is general and can be adapted to other problems related to sub- or super-stabilizability. MSC:26E60.

Sharp power mean bounds for Seiffert mean

In this paper, we find the greatest value p = log2/(log π − log 2) = 1.53 … and the least value q = 5/3 = 1.66 … such that the double inequality M p (a, b) 0 with a ≠ b. Here, M p (a, b) and T (a, b) are the p-th power and Seiffert means of two positive numbers a and b, respectively.

Monotonicity of the Ratio of the Power and Second Seiffert Means with Applications

We present the necessary and sufficient condition for the monotonicity of the ratio of the power and second Seiffert means. As applications, we get the sharp upper and lower bounds for the second Seiffert mean in terms of the power mean.

Sharp bounds for Seiffert mean in terms of weighted power means of arithmetic mean and geometric mean

For a,b > 0 with a = b , let P = (a− b)/(4arctana/b−π) , A = (a+ b)/2 , G = √ ab denote the Seiffert mean, arithmetic mean, geometric mean of a and b , respectively. In this paper, we present new sharp bounds for Seiffert P in terms of weighted power means of arithmetic mean A and geometric mean G : ( 2 3 A p1 + 3 G p1 )1/p1 < P < ( 2 3 A p2 + 3 G p2 )1/p2 , where p1 = 4/5 and p2 = logπ/2 (3/2) are the best possible constants. Moreover, our sharp bounds for P are compared with other known ones, which yields a chain of inequalities involving Seiffert mean P . Mathematics subject classification (2010): Primary 26E60, 26D05; secondary 33B10.

Sharp bounds for Seiffert mean in terms of arithmetic and geometric means

Sharp bounds for Seiffert mean in terms of arithmetic and geometric means

Optimal Lower Generalized Logarithmic Mean Bound for the Seiffert Mean

We present the greatest valuepsuch that the inequalityP(a,b)&gt;Lp(a,b)holds for alla,b&gt;0witha≠b, whereP(a,b)andLp(a,b)denote the Seiffert andpth generalized logarithmic means ofaandb, respectively.

Optimal Two Parameter Bounds for the Seiffert Mean

We obtain sharp bounds for the Seiffert mean in terms of a two parameter family of means. Our results generalize and extend the recent bounds presented in the Journal of Inequalities and Applications (2012) and Abstract and Applied Analysis (2012).

The optimal generalized logarithmic mean bounds for Seiffert's mean

For p ∈R, the generalized logarithmic mean Lp(a,b) and Seiffert's mean T(a,b) of two positive real numbers a and b are defined in (1.1) and (1.2) below respectively. In this paper, we find the greatest p and least q such that the double-inequality Lp(a,b) <T(a,b) <Lq(a,b) holds for all a,b > 0 and a ≠ b.

Sharp bounds for Seiffert mean in terms of root mean square

We find the greatest value α and least value β in (1/ 2,1) such that the double inequality S ( α a + ( 1 - α ) b , α b + ( 1 - α ) a ) <T ( a , b ) <S ( β a + ( 1 - β ) b , β b + ( 1 - β ) a ) holds for all a,b > 0 with a ≠ b. Here, T(a, b) = (a-b)/[2 arctan((a-b)/(a + b))] and S(a, b) = [(a2 + b2)/2]1/ 2are the Seiffert mean and root mean square of a and b, respectively.2010 Mathematics Subject Classification: 26E60.

Sharp Bounds for Seiffert Mean in Terms of Contraharmonic Mean

We find the greatest value α and the least value β in (1/2, 1) such that the double inequality C(αa + (1 − α)b, αb + (1 − α)a) &lt; T(a, b) &lt; C(βa + (1 − β)b, βb + (1 − β)a) holds for all a, b &gt; 0 with a ≠ b. Here, T(a, b) = (a − b)/[2 arctan((a − b)/(a + b))] and C(a, b) = (a2 + b2)/(a + b) are the Seiffert and contraharmonic means of a and b, respectively.

Two sharp double inequalities for Seiffert mean

In this paper, we establish two new inequalities between the root-square, arithmetic, and Seiffert means.The achieved results are inspired by the paper of Seiffert (Die Wurzel, 29, 221-222, 1995), and the methods from Chu et al. (J. Math. Inequal., 4, 581-586, 2010). The inequalities we obtained improve the existing corresponding results and, in some sense, are optimal.Mathematics Subject Classification (2010): 26E60.

The Optimal Convex Combination Bounds for Seiffert's Mean

We derive some optimal convex combination bounds related to Seiffert's mean. We find the greatest values , and the least values , such that the double inequalities and hold for all with . Here, , , , and denote the contraharmonic, geometric, harmonic, and Seiffert's means of two positive numbers and , respectively.

Sharp Generalized Seiffert Mean Bounds for Toader Mean

For p ∈ [0,1], the generalized Seiffert mean of two positive numbers a and b is defined by Sp(a, b) = p(a − b)/arctan[2p(a − b)/(a + b)], 0 &lt; p ≤ 1, a ≠ b; (a + b)/2, p = 0, a ≠ b; a, a = b. In this paper, we find the greatest value α and least value β such that the double inequality Sα(a, b) &lt; T(a, b) &lt; Sβ(a, b) holds for all a, b &gt; 0 with a ≠ b, and give new bounds for the complete elliptic integrals of the second kind. Here, denotes the Toader mean of two positive numbers a and b.