## Inequalities and monotonicity properties for the gamma function.(English)Zbl 0985.33003

After an extensive survey of previous results by the authors and others, they offer the inequalities $\Gamma[1+x+(1/x)]/2\leq \Gamma(1+x)\Gamma[1+(1/x)]<\Gamma[1+x+(1/x)]$ $$(x>0;$$ equality on the left at $$x=1$$), counter examples to the left inequality for more terms and factors, and proof of a weakened form of that generalization.

### MSC:

 33B15 Gamma, beta and polygamma functions
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### References:

 [1] Abramowitz, M.; Stegun (Eds.), I.A., Handbook of mathematical functions, (1964), National Bureau of Standards Washington, DC [2] Alzer, H., Some gamma function inequalities, Math. comp., 60, 337-346, (1993) · Zbl 0802.33001 [3] Alzer, H., On some inequalities for the gamma and psi functions, Math. comp., 66, 373-389, (1997) · Zbl 0854.33001 [4] Alzer, H., On Bernstein’s inequality for ultraspherical polynomials, Arch. math., 69, 487-490, (1997) · Zbl 0912.33004 [5] Alzer, H., A harmonic Mean inequality for the gamma function, J. comput. appl. math., 87, 195-198, (1997) · Zbl 0888.33001 [6] Bustoz, J.; Ismail, M.E.M., On gamma function inequalities, Math. comp., 47, 659-667, (1986) · Zbl 0607.33002 [7] Durand, L., Nicholson-type integrals for products of Gegenbauer functions and related topics, (), 353-374 [8] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher transcendental functions, vol. 1, (1953), McGraw-Hill Book Company, Inc New York, Toronto, London · Zbl 0052.29502 [9] Gautschi, W., Some elementary inequalities relating to the gamma and incomplete gamma function, J. math. phys., 38, 77-81, (1959) · Zbl 0094.04104 [10] Gautschi, W., A harmonic Mean inequality for the gamma function, SIAM J. math. anal., 5, 278-281, (1974) · Zbl 0239.33002 [11] Gautschi, W., Some Mean value inequalities for the gamma function, SIAM J. math. anal., 5, 282-292, (1974) · Zbl 0239.33003 [12] Gautschi, W., The incomplete gamma functions Since Tricomi, Atti convegni lincei roma (Proceedings of international meeting, Tricomi’s ideas and contemporary applied mathematics, Rome 28-29/11/1997 and Turin 1-2/12/1997), 147, 203-237, (1998) · Zbl 0965.33001 [13] Giordano, C.; Laforgia, A.; Pečarić, J., Unified treatment of gautschi – kershaw type inequalities for the gamma function, J. comput. appl. math., 99, 166-175, (1998) · Zbl 0932.33001 [14] Giordano, C.; Laforgia, A.; Pečarić, J., Monotonicity properties for some functions involving the ratio of two gamma functions, (), 35-42 [15] Ismail, M.E.H.; Muldoon, M.E., Inequalities and monotonicity properties for gamma and q-gamma functions, (), 309-323 · Zbl 0819.33001 [16] H.H. Kairies, An inequality for Krull solutions of a certain difference equation, in: E.F. Beckenbach, W. Walter (Eds.), ISNM General Inequalities, Vol. 3, Birkhäuser, Basel, 1983, pp. 277-280. · Zbl 0517.39001 [17] Kershaw, D., Some extensions of W. Gautschi’s inequalities for the gamma function, Math. comp., 41, 164, 607-611, (1983) · Zbl 0536.33002 [18] Laforgia, A., Further inequalities for the gamma function, Math. comp., 42, 166, 597-600, (1984) · Zbl 0536.33003 [19] Laforgia, A., A simple proof of the Bernstein inequality for ultraspherical polynomials, Boll. un. mat. ital., A (7) 6, 267-269, (1992) · Zbl 0753.33006 [20] Laforgia, A.; Sismondi, S., A geometric Mean inequality for the gamma function, Boll. un. mat. ital. A, (7) 3, 339-342, (1989) · Zbl 0686.33001 [21] Lorch, L., Inequalities for ultraspherical polynomials and the gamma function, J. approx. theory, 40, 115-120, (1984) · Zbl 0532.33007 [22] Merkle, M., Logarithmic convexity and inequalities for the gamma function, J. math. anal. appl., 203, 369-380, (1995) · Zbl 0860.33001 [23] Muldoon, M.E., Some monotonicity properties and characterizations of the gamma function, Aequationes math., 18, 54-63, (1978) · Zbl 0386.33001 [24] Palumbo, B., A generalization of some inequalities for the gamma function, J. comput. appl. math., 88, 2, 255-268, (1998) · Zbl 0902.33002 [25] G. Szegö, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ., 4th Edition, Vol. 23, Providence, RI, 1975.
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