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In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the functionis strictly increasing from (0, ∞) onto (1, 1860/343). This not only yields some known and new inequalities for the gamma function, but also gives some completely monotonic functions related to the gamma function.MSC: Primary 33B15; 26A48; secondary 26D15; 26A51

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In this paper, we present some new extensions of Hölder-type inequalities on time scales via diamond-α integral. Moreover, the obtained results are used to generalize Minkowski's inequality and Beckenbach-Dresher's inequality on time scales.

In this paper, we investigate the monotonicity pattern of the functionon (0,1) for a 1 and resolve an open problem. From which we prove that the double inequalityholds for x ∈ (0,1) if and only if 0 < a (1 − γ) / (2γ − 1) and b π 2 − 6γ / 18 − 12γ − π 2 , while the double inequalityholds for x ∈ (0,1) if and only if a (1 − γ) / (2γ − 1) and 0 < b 6γ/(π 2 − 12γ) , where γ = 0.577 ... denotes Euler-Mascheroni's constant. These greatly improve some existing results.Mathematics subject classification (2010): Primary 33B15, 26A48, Secondary 26D15.

Let f and g be both continuous functions on (0, ∞) with g (t) > 0 for t ∈ (0, ∞) and let F (x) = L (f ), G (x) = L (g) be respectively the Laplace transforms of f and g converging for x > 0. We prove that if there is a t * ∈ (0, ∞) such that f /g is strictly increasing on (0, t * ) and strictly decreasing on (t * , ∞), then the ratio F/G is decreasing on (0, ∞) if and only if ∞ 0 e −x cosh t cosh (vt) dt, 2010 Mathematics Subject Classification. Primary 44A10, 26A48; Secondary 33B15, 33C10. Key words and phrases. Laplace transform, monotonicity rule, psi function, modified Bessel functions of the second kind.This paper is in final form and no version of it will be submitted for publication elsewhere.

Abstract. In this paper, we present some new improvements of generalized Hölder's inequalities, and then we obtain a new refinement of Minkowski inequality. Moreover, the obtained results are used to improve Chung's inequality and Beckenbach-type inequality proposed by Wang.Mathematics subject classification (2010): 26D15, 26D10.