Science and Education Publishing Turkish Journal of Analysis and Number Theory 2 4 2014 09 08 A New Proof of an Inequality for the Logarithm of the Gamma Function and Its Sharpness 147 151 EN Mansour Mahmoud Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia TJANT2014248 10.12691/tjant-2-4-8 2014 07 03 2014 09 01 2014 09 08 In the paper, the author shows that the partial sums are alternatively larger and smaller than the generalized Euler's harmonic numbers with sharp bounds, where is the Euler's constant, are the Bernoulli numbers and is the digamma function.