Chao-Ping Chen

Some properties of functions related to the gamma and psi functions

In this paper, several properties associated with the complete monotonicity, the strongly complete monotonicity and the logarithmically complete monotonicity of functions related to the gamma and psi functions are obtained. Relevant connections of the results presented here with those derived in earlier works are also pointed out.

Sharp Cusa and Becker-Stark inequalities

AbstractWe determine the best possible constants θ,ϑ,α and β such that the inequalities 2+cosx3θ<sinxx<2+cosx3ϑ and π2π2-4x2α<tanxx<π2π2-4x2β are valid for 0 < × < π/ 2. Our results sharpen inequalities presented by Cusa, Becker and Stark.Mathematics Subject Classification (2000): 26D05.

Sharpness of Wilker and Huygens type inequalities

We present an elementary proof of Wilkers inequality involving trigonometric functions, and establish sharp Wilker and Huygens type inequalities.Mathematics Subject Classification 2010: 26D05.

Wilker- and Huygens-type inequalities and solution to Oppenheim’s problem

We establish Wilker- and Huygens-type inequalities for inverse trigonometric and inverse hyperbolic functions. We also provide a laconic proof to Oppenheim’s problem associated with inequalities involving the sine and cosine functions.

Some inequalities and monotonicity properties associated with the gamma and psi functions and the Barnes G-function

In this paper, several properties associated with inequalities and the logarithmically complete monotonicity of functions related to the gamma and psi functions and the Barnes G-function are obtained. Relevant connections of the results presented here with those derived in earlier works are also pointed out.

Notes on double inequalities of Mathieu's series

Using the integral expression of Mathieus series and some integral and analytic inequalities involving periodic functions and the generating function of Bernoulli numbers, we present several new inequalities and estimates for Mathieus series and generalize Mathieus series. Two open problems are proposed.

A class of two-sided inequalities involving the psi and polygamma functions

A class of two-sided inequalities involving the psi and polygamma functions is presented. These inequalities provide new bounds for the psi function in terms of the polygamma functions.

Note on weighted Carleman-type inequality

A double inequality involving the constant e is proved by using an inequality between the logarithmic mean and arithmetic mean. As an application, we generalize the weighted Carleman-type inequality.

New representations for the Lugo and Euler–Mascheroni constants

Abstract Lugo’s constant L given by L = − 1 2 − γ + ln 2 is defined as the limit of the sequence ( L n ) n ∈ N defined by L n : = ∑ i = 1 n ∑ j = 1 n 1 i + j − ( 2 ln 2 ) n + ln n ( n ∈ N ) as n → ∞ , N being the set of positive integers. In this paper, we establish new analytical representations for the Euler–Mascheroni constant γ in terms of the psi function. We also give the bounds of L − L n and present a new sequence which converges to Lugo’s constant L .

Another proof of monotonicity for extended mean values

We provide another proof of monotonicity for the extended mean values. Stolarsky defined in (5) the extended mean values E(r, s; x, y) by