Schwab-Borchardt Mean Research Articles

INEQUALITIES INVOLVING GENERALIZED TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS

Trigonometric and Hyperbolic inequalities, which have been obtained by C. Huygens[5], D. S. Mitrinovic[10] and many more, have attracted attention of several researchers. We oer several renements and generalization of few trigonometric and hyperbolic inequalities involving tangent function, cotangent function, sine function, secant function and cosecant function. The established results are obtained with the aid of the Schwab-Borchardt mean.

On a class of new means including the generalized Schwab-Borchardt mean

The so-called Schwab-Borchardt mean plays an important role in the theory of (bivariate) means. It includes a lot of standard means, such as the logarithmic mean, the first and second Seiffert means and the Neuman-Sándor mean. In this paper, we investigate an approach which allows us to construct a class of new means. Such class includes the (generalized) Schwab-Borchardt mean and other old/new means derived as well.

On thep-Version of the Schwab-Borchardt Mean II

This paper deals with thep-version of the Schwab-Borchardt mean. Lower and upper bounds for this mean, expressed in terms of the weighted geometric and arithmetic means of its variables, are obtained. Applications to four bivariate means, introduced earlier by the author of this paper, are included.

Optimal bounds for Neuman means in terms of geometric, arithmetic and quadratic means

In this paper, we present sharp bounds for the two Neuman means S H A and S C A derived from the Schwab-Borchardt mean in terms of convex combinations of either the weighted arithmetic and geometric means or the weighted arithmetic and quadratic means, and the mean generated either by the geometric or by the quadratic mean.MSC:26E60.

On the Inequalities for the Generalized Trigonometric Functions

This paper deals with Huygens-type and Wilker-type inequalities for the generalized trigonometric functions of P. Lindqvist. A major mathematical tool used in this work is a generalized version of the Schwab-Borchardt mean introduced recently by the author of this work.

On some means derived from the Schwab-Borchardt mean

Bivariate means defined as the Schwab-Borchardt mean of two bivariate means are investigated. Explicit formulas for those means are obtained. It is demonstrated that they inter- polate inequalities connecting the well known bivariate means. Optimal bounds for the means under discussion are also obtained. The bounding quantities are convex combinations of the generating means.

On thep-Version of the Schwab-Borchardt Mean

This paper deals with a one-parameter generalization of the Schwab-Borchardt mean. The new mean is defined in terms of the inverse functions of the generalized trigonometric and generalized hyperbolic functions. The four new bivariate means are introduced as particular cases of thep-version of the Schwab-Borchardt mean. For the particular value of the parameterp, these means become either the classical logarithmic mean or the Seiffert means or the Neuman-Sándor mean. Wilker- and Huygens-type inequalities involving inverse functions of the generalized trigonometric and the generalized hyperbolic functions are also established.

On some means derived from the Schwab-Borchardt mean II

On some means derived from the Schwab-Borchardt mean II

Three families of two-parameter means constructed by trigonometric functions

In this paper, we establish three families of trigonometric functions with two parameters and prove their monotonicity and bivariate log-convexity. Based on them, three two-parameter families of means involving trigonometric functions, which include Schwab-Borchardt mean, the first and second Seiffert means, Sándor’s mean and many other new means, are defined. Their properties are given and some new inequalities for these means are proved. Lastly, two families of two-parameter hyperbolic means, which similarly contain many new means, are also presented without proofs.MSC: 26E60, 26D05, 33B10, 26A48.

On a new bivariate mean

A new bivariate mean is introduced and studied. The mean under discussion is defined as the degenerate case of the completely symmetric elliptic integral of the second kind. Its relationship with the Schwab–Borchardt mean is established. Four particular cases of the mean in question are discussed. Sharp bounds and the Ky Fan inequalities for the quadruple of new means are established.

Optimal Bounds for Neuman Means in Terms of Harmonic and Contraharmonic Means

Fora,b>0witha≠b, the Schwab-Borchardt meanSB(a,b)is defined asSB(a,b)={b2-a2/cos-1(a/b) ifa<b,a2-b2/cosh-1(a/b) ifa>b. In this paper, we find the greatest values ofα1andα2and the least values ofβ1andβ2in[0,1/2]such thatH(α1a+(1-α1)b,α1b+(1-α1)a)<SAH(a,b)<H(β1a+(1-β1)b,β1b+(1-β1)a)andH(α2a+(1-α2)b,α2b+(1-α2)a)<SHA(a,b)<H(β2a+(1-β2)b,β2b+(1-β2)a). Similarly, we also find the greatest values ofα3andα4and the least values ofβ3andβ4in[1/2,1]such thatC(α3a+(1-α3)b,α3b+(1-α3)a)<SCA(a,b)<C(β3a+(1-β3)b,β3b+(1-β3)a)andC(α4a+(1-α4)b,α4b+(1-α4)a)<SAC(a,b)<C(β4a+(1-β4)b,β4b+(1-β4)a). Here,H(a,b)=2ab/(a+b),A(a,b)=(a+b)/2, andC(a,b)=(a2+b2)/(a+b)are the harmonic, arithmetic, and contraharmonic means, respectively, andSHA(a,b)=SB(H,A),SAH(a,b)=SB(A,H),SCA(a,b)=SB(C,A), andSAC(a,b)=SB(A,C)are four Neuman means derived from the Schwab-Borchardt mean.

Inequalities for weighted sums of powers and their applications

Two inequalities for weighted sums of powers are established. Applications to Jacobian elliptic functions and Legendre’s elliptic integrals of the first kind are presented. Some known and new inequalities for circular and hyperbolic functions are obtained. Applications to certain iterative means including Gauss’ arithmetic-geometric mean and Schwab-Borchardt mean are included. Mathematics subject classification (2010): Primary: 26D15, 33E05, 26D05, 26D07.

Inequalities for the Schwab-Borchardt mean and their applications

Inequalities for the Schwab-Borchardt mean are obtained. They contain known results for the trigonometric and hyperbolic functions including those obtained by J. Wilker (15) and C. Huygens (5). The main results of this paper can also be utilized to obtain new inequalities for some bivariate means including the logarithmic mean and two means introduced by Seiffert.

Total Positivity of Mean Values and Hypergeometric Functions

The weighted power mean of two positive variables is strictly totally positive (STP) if its order t satisfies $ - \infty < t < 0$, and its reciprocal is STP if $0 < t < \infty $. The reciprocals of the logarithmic mean, Gauss’s arithmetic-geometric mean, and the Schwab–Borchardt mean are STP. The hypergeometric R-function $R_{ - \alpha } (\beta ,\beta ':x,y),x,y > 0$, which is equivalent to ${}_2 F_1 $ with argument ${{1 - x} / y}$, is STP if $\alpha ,\beta \beta '$, and $\beta + \beta ' - \alpha $ are positive. With weaker restrictions this function is represented in a new way as a convolution. Higher order positivity is discussed for some other hypergeometric functions, including incomplete elliptic integrals.