Applicable Analysis and Discrete Mathematics 2011 Volume 5, Issue 1, Pages: 87-92
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On a binomial coefficient and a product of prime numbers

Alzer Horst A., Šandor Jozsef J.B.

Let Pn be the n-th prime number. We prove the following double-inequality. For all integers k≥5 we have exp[k(c0≥loglogk)]≥ k2 k/p1≥p2≥...≥pk ≥ exp[k(c1≥loglogk)] with the best possible constants c0 = 1/5 log23 + loglog5=1:10298… and c1 = 1/192log(36864/192)+loglog 192≥1/192log(p1≥p2≥…≥p192)=2.04287... This reffines a result published by Gupta and Khare in 1977.

Keywords: Prime numbers, binomial coefficients, Chebychev's function, inequalities, monotonicity