REFINEMENTS AND SHARPENINGS OF SOME DOUBLE INEQUALITIES FOR BOUNDING THE GAMMA FUNCTION

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1 REFINEMENTS AND SHARPENINGS OF SOME DOUBLE INEQUALITIES FOR BOUNDING THE GAMMA FUNCTION BAI-NI GUO YING-JIE ZHANG School of Mathematics and Informatics Department of Mathematics Henan Polytechnic University Jiaozuo University Jiaozuo City, Henan Province Jiaozuo City, Henan Province , China , China Received: 1 May, 007 Accepted: January, 008 FENG QI Research Institute of Mathematical Inequality Theory Henan Polytechnic University Jiaozuo City, Henan Province, , China qifeng618@hotmail.com URL: Page 1 of 9 Communicated by: P. Cerone 000 AMS Sub. Class.: Primary 33B15; Secondary 6D07. Key words: Inequality, Refinement, Sharpening, Generalization, Kečlić-Vasić-Alzer s double inequalities.

2 Abstract: In this paper, some sharp inequalities for bounding the gamma function Γ(x) and the ratio of two gamma functions are established. From these, several known results are recovered, refined, extended and generalized simply and elegantly. Acknowledgements: The authors would like to express heartily many thanks to the anonymous referee(s) for careful corrections to the original version of this manuscript. The first and third authors were supported in part by the NSF of Henan University, China. The third author was also supported in part by the China Scholarship Council in 008. Page of 9

3 In [4], it was proved that the function (1) f(x) = ln Γ(x + 1) x ln x is strictly increasing from (1, ) onto (1 γ, 1), where γ is Euler-Mascheroni s constant. In particular, for x (1, ), () x (1 γ)x 1 < Γ(x) < x x 1. In [1, Theorem ], inequality () was extended and sharpened: If x (0, 1), then (3) x α(x 1) γ < Γ(x) < x β(x 1) γ with the best possible constants α = 1 γ and β = 1 γ). If x (1, ), then 6 ( inequality (3) holds with the best possible constants α = 1 π γ) and β = 1. 6 In [8], by using the convolution theorem for Laplace transforms and other techniques, inequalities () and (3) were refined: The double inequality (4) x x γ xx 1/ < Γ(x) < ex 1 e x 1 holds for x > 1 and the constants γ and 1 are the best possible. For 0 < x < 1, the left-hand inequality in (4) still holds, but the right-hand inequality in (4) reverses. Remark 1. The double inequality (4) can be verified simply as follows: In [3], the function (5) θ(x) = x[ln x ψ(x)] was proved to be decreasing and convex in (0, ) with θ(1) = γ and two limits lim x 0 + θ(x) = 1 and lim x θ(x) = 1. Since the function g α(x) = ex Γ(x) x x α for x > ( π Page 3 of 9

4 0 satisfies xg α (x) g α(x) = x[ψ(x) ln x] + α, it increases for α 1, decreases for α 1, and has a unique minimum for 1 < α < 1 in (0, ). This implies that the function g α (x) decreases in (0, x 0 ) and increases in (x 0, ) for α = x 0 [ln x 0 ψ(x 0 )] and all x 0 (0, ). Hence, taking x 0 = 1 yields that α = γ and g γ (x) decreases in (0, 1) and increases in (1, ), and taking α = 1 gives that the function g 1/(x) is decreasing in (0, ). By virtue of g α (1) = e, the double inequality (4) follows. The first main result of this paper is the following theorem which can be regarded as a generalization of inequalities (), (3) and (4). e x Γ(x) Theorem 1. Let a be a positive number. Then the function is decreasing x x a[ln a ψ(a)] in (0, a] and increasing in [a, ), and the function ex Γ(x) in (0, ) is increasing if x x b and only if b 1 and decreasing if and only if b 1. Proof. This follows from careful observation of the arguments in Remark 1. For a > 0 and b > 0 with a b, the mean ( b b (6) I(a, b) = 1 e a a ) 1/(b a) is called the identric or exponential mean. See [9] and related references therein. As direct consequences of Theorem 1, several sharp inequalities related to the identric mean and the ratio of gamma functions are established as follows. Theorem. For y > x 1, (7) Γ(x) Γ(y) < xx γ y y γ ey x or [I(x, y)] y x < If 1 y > x > 0, inequality (7) reverses. ( ) γ y Γ(y) x Γ(x). Page 4 of 9

5 For y > x > 0, inequality (8) Γ(x) Γ(y) < xx b y y b ey x or [I(x, y)] y x < ( ) b y Γ(y) x Γ(x) holds if and only if b 1. The reversed inequality (8) is valid if and only if b 1. Proof. Letting a = 1 in Theorem 1 gives that the function ex Γ(x) is decreasing in x x γ (0, 1] and increasing in [1, ). Thus, for y > x 1, (9) e x Γ(x) x x γ < ey Γ(y) y y γ. Rearranging (9) leads to the inequalities in (7). The rest of the proofs are similar, so we shall omit them. Remark. The inequalities in (7) and (8) have been obtained in [7] and [, Theorem 4]. However, Theorem provides an alternative and concise proof of Kečlić- Vasić-Alzer s double inequalities in [, 7]. In [5, 6], several new inequalities similar to (7) and (8) were presented. The third main results of this paper are refinements and sharpenings of the double inequalities (), (3) and (4), which are stated below. Theorem 3. The function (10) h(x) = e x Γ(x) xx[1 ln x+ψ(x)] in (0, ) has a unique maximum e at x = 1, with the limits Page 5 of 9 (11) lim h(x) = 1 and lim h(x) = π. x 0 + x

6 Consequently, sharp double inequalities (1) in (0, 1] and (13) in [1, ) are valid. x[1 ln x+ψ(x)] x e x π x x[1 ln x+ψ(x)] e x Proof. Direct calculation yields < Γ(x) xx[1 ln x+ψ(x)] e x 1 < Γ(x) xx[1 ln x+ψ(x)] e x 1 (14) h (x) = [ln x ψ(x) xψ x x x[ln x ψ(x) 1] Γ(x) ln x. Since the factor xψ (x)+ψ(x) ln x 1 = θ (x) and θ(x) is decreasing in (0, ), the function h(x) has a unique maximum e at x = 1. The second limit in (11) follows from standard arguments by using the following two well known formulas: As x, (15) (16) ln Γ(x) = ( x 1 Direct computation gives ) ln x x + ln(π) ψ(x) = ln x 1 x 1 1x + O + 1 ( 1 1x + O x ( ) 1. (17) lim ln h(x) = lim x 0 + x 0 +[ln Γ(x) xψ(x) ln x] = 0 by utilizing the following two well known formulas [ ( (18) ln Γ(x) = ln x + γx + ln 1 + x ) x ] k k k=1 x ), Page 6 of 9

7 and (19) ψ(x) = γ + for x > 0. The proof is complete. k=0 ( 1 k ) x + k Remark 3. The graph in Figure 1 plotted by MATHEMATICA 5. shows that the left Figure 1: Graph of xx γ e x π x x[1 ln x+ψ(x)] e x in (1, 5) hand sides in double inequalities (4) and (13) for x > 1 do not include each other and that the lower bound in (13) is better than the one in (4) when x > 1 is large enough. As discussed in Remark 1, the double inequality 1 (0, ) clearly holds. Therefore, the upper bounds in (1) and (13) are better than the corresponding one in (4). < x[ln x ψ(x)] < 1 in Page 7 of 9

8 Theorem 4. Inequality (0) I(x, y) < { } x x[ln x ψ(x)] 1/(x y) Γ(x) y y[ln y ψ(y)] Γ(y) holds true for x 1 and y 1 with x y. If 0 < x 1 and 0 < y 1 with x y, inequality (0) is reversed. Proof. From Theorem 3, it is clear that the function h(x) is decreasing in [1, ) and increasing in (0, 1]. A similar argument to the proof of Theorem straightforwardly leads to inequality (0) and its reversed version. Remark 4. The inequality (0) is better than those in (7), since the function t[ψ(t) ln t] γ (1) q(t) t is decreasing in (0, ) with q(1) = 1 and lim t 0 + q(t) =, which is shown by the graph of q(t), plotted by MATHEMATICA 5.. It is conjectured that the function q(t) is logarithmically completely monotonic in (0, ). Page 8 of 9

9 References [1] H. ALZER, Inequalities for the gamma function, Proc. Amer. Math. Soc., 18(1) (1999), [] H. ALZER, Some gamma function inequalities, Math. Comp., 60 (1993) [3] G.D. ANDERSON, R.W. BARNARD, K.C. RICHARDS, M.K. VAMANA- MURTHY AND M. VUORINEN, Inequalities for zero-balanced hypergeometric functions, Trans. Amer. Math. Soc., 347(5) (1995), [4] G.D. ANDERSON AND S.-L. QIU, A monotonicity property of the gamma function, Proc. Amer. Math. Soc., 15 (1997), [5] C.-P. CHEN AND F. QI, Logarithmically completely monotonic functions relating to the gamma function, J. Math. Anal. Appl., 31 (006), [6] S. GUO, F. QI, AND H.M. SRIVASTAVA, Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic, Integral Transforms Spec. Funct., 18(11) (007), [7] J. D. KEČLIĆ AND P. M. VASIĆ, Some inequalities for the gamma function, Publ. Inst. Math. (Beograd) (N. S.), 11 (1971), [8] X. LI AND CH.-P. CHEN, Inequalities for the gamma function, J. Inequal. Pure Appl. Math., 8(1) (007), Art. 8. [ONLINE: article.php?sid=84]. [9] F. QI, The extended mean values: Definition, properties, monotonicities, comparison, convexities, generalizations, and applications, Cubo Mat. Educ. 5(3) (003), RGMIA Res. Rep. Coll., 5(1) (00), Art. 5, [ONLINE: Page 9 of 9