12,365 research outputs found
Some sharp inequalities involving Seiffert and other means and their concise proofs
In the paper, by establishing the monotonicity of some functions involving
the sine and cosine functions, the authors provide concise proofs of some known
inequalities and find some new sharp inequalities involving the Seiffert,
contra-harmonic, centroidal, arithmetic, geometric, harmonic, and root-square
means of two positive real numbers and with .Comment: 10 page
Geometric convexity of the generalized sine and the generalized hyperbolic sine
In the paper, the authors prove that the generalized sine function
and the generalized hyperbolic sine function
are geometrically concave and geometrically convex, respectively. Consequently,
the authors verify a conjecture posed in the paper "B. A. Bhayo and M.
Vuorinen, On generalized trigonometric functions with two parameters, J.
Approx. Theory 164 (2012), no.~10, 1415\nobreakdash--1426; Available online at
\url{http://dx.doi.org/10.1016/j.jat.2012.06.003}".Comment: 5 page
Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean
The authors find the greatest value λ and the least value μ, such that the double inequality C¯(λa+(1-λb),λb+(1-λ)a)0 with a≠b, where C¯(a,b)=2(a2+ab+b2)/3(a+b), A(a,b)=(a+b)/2, and Ta,b=2/π∫0π/2a2cos2θ+b2sin2θdθ denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers a and b
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