Equivalent conditions of a Hardy-type integral inequality related to the extended Riemann zeta function
Abstract. By the use of techniques of real analysis and weight functions, we obtain two lemmas and build a few equivalent conditions of a Hardy-type integral inequality with a non-homogeneous kernel, related to a parameter where the constant factor is expressed in terms of the extended Riemann zeta function. Meanwhile, a few equivalent conditions for two kinds of Hardytype integral inequalities with the homogeneous kernel are deduced. We also consider the operator expressions.
On a more accurate reverse Hilbert-type inequality in the whole plane
In the present paper, using weight coefficients and applying Hermite-Hadamard’s inequality, we derive a new, more accurate reverse Hilbert-type inequality in the whole plane with multi-parameters involving the cosine and natural logarithm functions. The corresponding constant factor is proved to be the best possible. We additionally consider some equivalent forms and a few particular inequalities. As an application, the obtained results are compared with some previously known results and we show that these new results are more accurate than the earlier ones. Mathematics subject classification (2010): 26D15, 47A05.
A Turán-type inequality for the gamma function
Abstract We prove that the following Turan-type inequality holds for Euler's gamma function. For all odd integers n ⩾ 1 and real numbers x > 0 we have α ⩽ Γ ( n − 1 ) ( x ) Γ ( n + 1 ) ( x ) − Γ ( n ) ( x ) 2 , with the best possible constant α = min 1.5 ⩽ x ⩽ 2 Γ ( x ) 2 ψ ′ ( x ) = 0.6359 … .