Improvements of the bounds for Ramanujan constant function
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In the article, we establish several inequalities for the Ramanujan constant function \(R(x)=-2\gamma-\psi(x)-\psi(1-x)\) on the interval \((0, 1/2]\), where \(\psi(x)\) is the classical psi function and \(\gamma=0.577215\cdots\) is the Euler-Mascheroni constant.
KeywordsRamanujan constant function gamma function psi function Euler-Mascheroni constant
It is well known that the gamma and psi functions have many applications in the areas of mathematics, physics, and engineering technology. Recently, the bounds for the gamma and psi functions have attracted the interest of many researchers. In particular, many remarkable inequalities for the psi function \(\psi(x)\) can be found in the literature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15].
The main purpose of this paper is to improve inequality (1.2).
In order to prove our main results we need several lemmas, which we present in this section.
(See , Section 3, in the proof of Theorem 5, pp. 2500-2502)
We clearly see that both the first and the second inequalities in (2.8) become to equations if \(x=\pi/2\). If \(x=0\), then the first inequality of (2.8) also holds and the second inequality of (2.8) becomes to equation.
3 Main results
The research was supported by the Natural Science Foundation of China under Grants 11371125, 61374086 and 11401191, and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.
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