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In the article, we establish several inequalities for the Ramanujan constant function R ( x ) = − 2 γ − ψ ( x ) − ψ ( 1 − x ) $R(x)=-2\gamma-\psi(x)-\psi(1-x)$ on the interval ( 0 , 1 / 2 ] $(0, 1/2]$ , where ψ ( x ) $\psi(x)$ is the classical psi function and γ = 0.577215 ⋯ $\gamma=0.577215\cdots$ is the Euler-Mascheroni constant

Improvements of the bounds for Ramanujan constant function