Horst Alzer A MEAN VALUE INEQUALITY FOR THE DIGAMMA FUNCTION

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1 Rendiconti Sem. Mat. Univ. Pol. Torino Vol. 75, 2 (207), 9 25 Horst Alzer A MEAN VALUE INEQUALITY FOR THE DIGAMMA FUNCTION Abstract. A recently published result states that for all ψ is greater than or equal to γ, that is, γ H,ψ 0. 0 the harmonic mean of and Here, ψ Γ Γ denotes the digamma function and γ is Euler s constant. We offer a proof for the following refinement: γh, H,ψ 0. Keywords: Digamma function, polygamma functions, harmonic mean, Euler s constant, inequality. Introduction and main result The classical harmonic mean of two real numbers a and b (which are not both equal to 0) is defined by 2ab H a,b a b. This mean value plays a role in various fields of mathematics and it also appears in physics, chemistry and computer science. Many inequalities for the harmonic and other means can be found in the monograph [0]. In 974, Gautschi [3] published a remarkable mean value inequality for Euler s gamma function. He showed that for all positive real numbers the harmonic mean of Γ and Γ is greater than or equal to, that is, () H Γ,Γ. 9

2 20 Horst Alzer Various generalizations, improvements and companions of () are given in [2], [3], [4], [5], [7], [8], [4], [5], [6], [7]. In a recent paper, Jameson and the author [9] proved a counterpart of () for the digamma function d d logγ Γ Γ. For all 0 we have (2) γ H,ψ, where the constant lower bound is best possible. Here, as usual, γ Euler s constant denotes The digamma function has interesting applications in numerous fields, like, for instance, the theory of special functions, statistics, mathematical physics and number theory. See [], [2], [8], [9], [2]. The following series and integral representations are valid for 0: γ k k k 0 e t t e t e t dt. ψ has a zero at (In what follows, we maintain this notation.) Moreover, ψ is strictly increasing and strictly concave on 0, and satisfies the limit relations lim 0 and lim log. These and many other properties of the ψ-function are given, for eample, in [], [20]. It is the aim of this note to offer a refinement of (2). We show that the following mean value inequality holds. THEOREM. For all positive real numbers we have γh, H,ψ. The sign of equality holds if and only if. In order to verify the Theorem we need several lemmas. They are collected in the net section. A proof of the Theorem is given in Section 3. The numerical values have been calculated via the computer program MAPLE 3.

3 A Mean Value Inequality 2 2. Lemmas In this section, we present properties of functions which are defined in terms of the ψ-function and its derivatives which are known as polygamma functions. We recall that n ψ n 0 for n N and 0. In particular, ψ is positive on 0,. Proofs for the first two lemmas can be found in [6]. LEMMA. Let k be an integer, c be a real number and (3) f c,k c ψ k. (i) f c,k is strictly decreasing on 0, if and only if c k. (ii) f c,k is strictly increasing on 0, if and only if c k. LEMMA 2. Let k be an integer. The function (4) g k ψ k ψ k is strictly increasing on 0, with lim g k k. LEMMA 3. The function (5) h ψ is positive on 0, 0 0, and negative on 0, 0. Proof. We define for 0, : Then, u ψ. u f, f,, where f, is given in (3). Since 0, we conclude from Lemma (i) that f, f,. It follows that u is strictly increasing on 0,. Net, let 0 0. Then, ψ u u 0 ψ 0.

4 22 Horst Alzer Hence, This leads to (6) Since we obtain (7) 0 ψ ψ 0. ψ 0 ψ. 0 ψ 0, ψ 0, so that (6) and (7) imply that h is positive on 0, 0. Since h h, we conclude that h is also positive on 0,. If 0, 0, then 0 and ψ 0. This reveals that h is negative on 0, 0. LEMMA 4. The function (8) v ψ 2 is strictly decreasing on 0, 0. γ Proof. Let 0, 0 and w g 2 f,, where f, and g are defined in (3) and (4), respectively. Then we obtain (9) v ψ 3 Applying Lemma (i) and Lemma 2 reveals that f, is decreasing on 0, g is positive and decreasing on 0,. This leads to w (0) w ψ 0 g 0 2 f, γ. and that Let 0 r s 0. Since ψ 3 is positive and increasing on 0, 0 and ψ is positive and decreasing on 0,, we obtain from (9) and (0): () v 2.07 We have ψ 3 γ 2.07 ψ 3 r ψ s y 0, and y 0.9, , so that () reveals that v is negative on 0, 0. γ y r,s, say.

5 A Mean Value Inequality Proof of the Theorem We define for 0: Since ψ γ, we obtain H H,ψ. γh, H γ. Thus, it remains to show that (2) γh, H for 0,. We consider two cases. Case Let h be the function defined in (5). Lemma 3 yields H 2 h 0 γh,. Case Using Lemma 3 we obtain that in order to prove (2) it suffices to show that the function z h γ is negative on 0,. We have (3) z v v, where v is given in (8). Since 0 0, we conclude from Lemma 4 that v v. Applying (3) reveals that z is strictly increasing on 0,. Thus, This completes the proof of the Theorem. z z 0. Acknowledgement. I thank the referee for helpful comments. References [] ABRAMOWITZ M. AND I.A. STEGUN (EDS.), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York 992. [2] ALZER H., A harmonic mean inequality for the gamma function, J. Comput. Appl. Math. 87 (997),

6 24 Horst Alzer [3] ALZER H., Inequalities for the gamma function, Proc. Amer. Math. Soc. 28 (999), [4] ALZER H., On a gamma function inequality of Gautschi, Proc. Edinb. Math. Soc. 45 (2002), [5] ALZER H., On Gautschi s harmonic mean inequality for the gamma function, J. Comput. Appl. Math. 57 (2003), [6] ALZER H., Mean-value inequalities for the polygamma functions, Aequat. Math. 6 (200), 5 6. [7] ALZER H., Inequalities involving Γ and Γ, J. Comput. Appl. Math. 92 (2006), [8] ALZER H., Gamma function inequalities, Numer. Algor. 49 (2008), [9] ALZER H. AND G. JAMESON, A harmonic mean inequality for the digamma function and related results, Rend. Sem. Mat. Univ. Padova 37 (207), [0] BULLEN P.S., D.S. MITRINOVIĆ, AND P.M. VASIĆ, Means and Their Inequalities, Reidel, Dordrecht 988. [] COFFEY M.C., On one-dimensional digamma and polygamma series related to the evaluation of Feynman diagrams, J. Comp. Appl. Math. 83 (2005), [2] DE DOELDER P.J., On some series containing ψ y and ψ y 2 for certain values of and y, J. Comput. Appl. Math. 37 (99), [3] GAUTSCHI W., A harmonic mean inequality for the gamma function, SIAM J. Math. Anal. 5 (974), [4] GAUTSCHI W., Some mean value inequalities for the gamma function, SIAM J. Math. Anal. 5 (974), [5] GIORDANO C. AND A. LAFORGIA, Inequalities and monotonicity properties for the gamma function, J. Comp. Appl. Math. 33 (200), [6] JAMESON G.J.O. AND T.P. JAMESON, An inequality for the gamma function conjectured by D. Kershaw, J. Math. Ineq. 6 (202), [7] LAFORGIA A. AND S. SISMONDI, A geometric mean inequality for the gamma function, Boll. Un. Mat. Ital. A (7) 3 (989), [8] MURTY M.R. AND N. SARADHA, Transcendental values of the digamma function, J. Number Th. 25 (2007), [9] OGREID O.M. AND P. OSLAND, Some infinite series related to Feynman diagrams, J. Comput. Appl. Math. 40 (2002),

7 A Mean Value Inequality 25 [20] OLVER F.W.J., D.W. LOZIER, R.F. BOISVERT, AND C.W. CLARK (EDS.), NIST Handbook of Mathematical Functions, Camb. Univ. Press, New York 200. [2] TEMME N.M., Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York 996. AMS Subject Classification: 33B5, 39B62 Horst ALZER Morsbacher Straße 0, 5545 Waldbröl, GERMANY Lavoro pervenuto in redazione il