Monotonicity of the ratio of modified Bessel functions of the first kind with applications.

Let [Formula: see text] with [Formula: see text] be the modified Bessel functions of the first kind of order . In this paper, we prove the monotonicity of the function [Formula: see text] on [Formula: see text] for different values of parameter with [Formula: see text]. As applications, we deduce some new Simpson-Spector-type inequalities for [Formula: see text] and derive a new type of bounds [Formula: see text] ([Formula: see text]) for [Formula: see text].

Sharp Smith's bounds for the gamma function.

Among various approximation formulas for the gamma function, Smith showed that [Formula: see text] which is a little-known but accurate and simple one. In this note, we prove that the function [Formula: see text] is strictly increasing and concave on [Formula: see text], which shows that Smith's approximation is just an upper one.

Complete monotonicity involving some ratios of gamma functions.

In this paper, by using the properties of an auxiliary function, we mainly present the necessary and sufficient conditions for various ratios constructed by gamma functions to be respectively completely and logarithmically completely monotonic. As consequences, these not only unify and improve certain known results including Qi's and Ismail's conclusions, but also can generate some new inequalities.

Sharp inequalities for tangent function with applications.

In the article, we present new bounds for the function [Formula: see text] on the interval [Formula: see text] and find sharp estimations for the Sine integral and the Catalan constant based on a new monotonicity criterion for the quotient of power series, which refine the Redheffer and Becker-Stark type inequalities for tangent function.

[Anterior chamber phakic intraocular lens implantation in highly myopic eyes].

Objective: To investigate the stability, predictability and safety of angle-supported phakic intraocular lens (IOL) implantation in high myopia.

Methods: Seventy-six myopic eyes of 40 patients treated with anterior chamber angle-supported phakic IOL (Phakic6H) implantation were followed. The average myopia diopter was (-15.