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‎Let ‎‎$‎g:‎\mathbb{R^+}‎‎‎\rightarrow‎‎\mathbb{R^+}‎$ ‎be a‎ ‎real‎ ‎function ‎such ‎that ‎‎$\log g(x)$ ‎is ‎concave and ‎$‎\displaystyle{\lim_{x\to\infty}}‎\frac{g(x+w)}{g(x)} = 1‎$ ‎for each ‎$‎w>0‎$‎‎‎‎.‎ My questions are:‎‎ ‎‎‎‎‎‎‎‎‎‎‎‎‎

(a) Is the following inequality known‏?

‎ ‎‎‎‎‎‎‎‎‎$‎0‎\leq ‎‎\gamma‎_g + ‎\log ‎g(1)\leq ‎‎\frac{g^‎\prime_{-}(1) + g^‎\prime_{+}(1) ‎}{2g(1)}‎‎‎‎$ ‎‎ ‎

‎where‎ ‎‎$\sum_{i=1}^n‎‎‎\frac{g^‎\prime_{-}(i) + g^‎\prime_{+}(i) ‎}{2g(i)} - \log g(n)‎‎\rightarrow‎‎ ‎‎\gamma‎_g‎$ ‎as ‎‎$‎n‎‎\rightarrow‎\infty‎$‎.‎ ‎

(b) ‎‎H‎ow ‎to ‎get ‎it?

‎please guide me.

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