Journal of Mathematical Analysis and Applications

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1 J. Math. Anal. Appl ) Contents lists available at ScienceDirect Journal of Mathematical Analysis and Applications A Turán-type inequality for the gamma function Horst Alzer a, Giovanni Felder b, a Morsbacher Str. 1, Waldbröl, Germany b Department of Mathematics, ETH Zurich, 892 Zurich, Switzerland article info abstract Article history: Received 12 October 27 Available online 27 September 28 Submitted by I. Podlubny We prove that the following Turán-type inequality holds for Euler s gamma function. For all odd integers n 1 and real numbers > wehave α Ɣ n 1) )Ɣ n+1) ) Ɣ n) ) 2, Keywords: Turán-type inequality Gamma function Psi function with the best possible constant α min Ɣ) 2 ψ ) Elsevier Inc. All rights reserved. 1. Introduction In 195, Turán 22] proved the inequality P n 1 )P n+1 ) P n ) 2 1 1, n 1, 2,...), P n denotes the Legendre polynomial of degree n. This inequality has attracted much attention, so that numerous inequalities of the same type were published for other special functions. In 1986, Csordas, Norfolk and Varga 7] proved a Turán-type inequality, which is a necessary condition for the validity of the famous Riemann hypothesis. Inequalities of Turán-type are studied in various fields, like, for eample, comple analysis, number theory, combinatorics, and theory of mean-values. Also, they have applications in statistics and control theory. We refer to 6,8,9,11,13 19,21] and the references given therein. In this paper we are concerned with a Turán-type inequality for Euler s gamma function Ɣ) e t t 1 dt > ). There eist many inequalities for this important function and its relatives see 12,2]), but inequalities involving higher order derivatives of the Ɣ-function are difficult to find in the literature. It is known that the Cauchy Schwarz inequality can be applied to obtain estimates for special functions; see, for eample, 1] and 14]. We find for odd integers n 1 and real numbers > : * Corresponding author. addresses: H.Alzer@gm.de H. Alzer), felder@math.ethz.ch G. Felder) X/$ see front matter 28 Elsevier Inc. All rights reserved. doi:1.116/j.jmaa

2 Ɣ n) ) 2 H. Alzer, G. Felder / J. Math. Anal. Appl ) ) 2 e t t 1 logt) n 1] 1/2 e t t 1 logt) n+1] 1/2 dt e t t 1 logt) n 1 dt e t t 1 logt) n+1 dt Ɣ n 1) )Ɣ n+1) ). As usual, Ɣ ) Ɣ.) Thus we have Ɣ n 1) )Ɣ n+1) ) Ɣ n) ) 2 n ). In what follows we maintain this notation.) Is it possible to replace the lower bound by a positive constant? It is our aim to give an affirmative answer to this question. More precisely we determine the largest real number α, which is independent of n and, such that we have α n ) >, n 1, 3, 5,...). 1.1) In the net section we collect some lemmas. They play an important role in the proof of our main result, given in Section 3. The numerical and algebraic computations have been carried out by MAPLE V, Release Lemmas The first five lemmas provide properties of the psi function, ψ Ɣ /Ɣ, and its derivatives. We denote by theonlypositivezeroofψ. Lemma 1. i) ψ is strictly increasing on, ). ii) ψ and ψ are positive and strictly decreasing on, ). iii) ψ is negative and strictly increasing on, ). Lemma 1 follows from the integral formula ψ n) ) 1) n+1 see, for instance, 1, p. 26]. e t t n dt >, n 1, 2,...), 2.1) 1 e t Lemma 2. i) The function ψ) is strictly decreasing on, r ] and strictly increasing on r, ),r ii) The functions 2 ψ ) and 3 ψ ) are strictly increasing on, ). A proof for part i) can be found in 3]. Part ii) is a special case of a more general monotonicity theorem, which is proved in 4]. Lemma 3. For all > we have <ψ )ψ ) ψ ) ) An application of 2.1) and of the Cauchy Schwarz inequality leads to 2.2). See 5] and 14] for corresponding inequalities involving higher derivatives. Lemma 4. For all integers n and > we have ψ n) + 1) ψ n) ) + 1) n n! n+1. This recurrence formula is given in 1, p. 26].

3 278 H. Alzer, G. Felder / J. Math. Anal. Appl ) Lemma 5. For all > we have log) 1 <ψ)<log), 1 <ψ ) and <ψ ). 2.3) 2.4) Proof. A proof for 2.3) can be found in 2]. Using the integral representations 2.1) and as well as n 1)! n e t t n 1 dt >, n 1, 2,...) e z 1 z > z ) we obtain for > : and ψ ) 1 ψ ) This proves 2.4). e t e t 1 + t 1 e t dt > e +1)t tet 1 t) 1 e t dt >. The net three lemmas present properties of n. Lemma 6. Let n 1 be an odd integer. Then, n is conve on, ). Proof. We set n 2k 1withk 1. Differentiation gives for > : 2k 1 ) Ɣ2k 2) )Ɣ 2k+2) ) Ɣ 2k) ) 2. Applying the Cauchy Schwarz inequality yields ). 2k 1 Remark. Let n 1 be an odd integer. Since n is nonnegative and conve on, ), we conclude that the following Schur-type functional inequality holds for all real numbers, y, z > : y) z) n ) + y )y z) n y) + z )z y) n z), see 23]. Lemma 7. For all > we have 1 )< 3 ). Proof. We have 3 ) 1 ) H)Ɣ) 2, H) ψ) 2 ψ ) + 2ψ) 3 ψ ) + 3ψ ) 3 + ψ )ψ ) + ψ )ψ) 4 2ψ)ψ )ψ ) ψ ) ψ ) 2. To prove that H is positive on, ) we distinguish four cases. Case 1. < 1/4.

4 H. Alzer, G. Felder / J. Math. Anal. Appl ) Applying Lemmas 1 and 3 gives H)>I) + J), I) 2ψ) 3 ψ ) 2ψ)ψ )ψ ) and J) 3ψ ) 3 ψ ). We have I) 2ψ)ψ )K ), K ) ψ) 2 ψ ). Using Lemmas 1 and 4 leads to K) ψ + 1) 2 + 2ψ + 1) + ψ + 1)<2ψ + 1) + ψ + 1) 2ψ5/4) ψ 1).4... This proves I)>. Furthermore, Lemma 1 yields J) ψ ) 3ψ ) 2 1 3ψ 1/4) Hence, J)>. Case 2. 1/4. Lemmas 1 and 3 give H) 3ψ ) 3 + ψ )ψ) 4 2ψ)ψ )ψ ) ψ ) ψ )M) 4, M) 3 2 ψ ) ] 2 + ψ) ] 4 2 ψ) ] 3 ψ ) ] 4. Let 1/4 r s. Applying Lemma 2 we obtain M) 3 r 2 ψ r) ] 2 + sψs) ] 4 2 rψr) ] s 3 ψ s) ] s 4 Nr, s), say. Since the numbers N.25,.46), N.46,.75), N.75, 1.18), N1.18, ) are positive, we conclude that M)>for 1/4, ]. This implies that H is positive on 1/4, ]. Case Using 2.2) leads to H)>ψ) 2 ψ ) + 2ψ) 3 ψ ) + 3ψ ) 3 + ψ )ψ) 4 2ψ)ψ )ψ ) ψ ). Let r s 5. Then we get from Lemma 1: H)>ψr) 2 ψ s) + 2ψs) 3 ψ r) + 3ψ s) 3 + ψ s)ψr) 4 2ψr)ψ s)ψ s) ψ r) Pr, s), say. We have P, 1.8)> and P k/1, k + 1)/1 ) > for k, 1,...,319. This implies that H)>for, 5].

5 28 H. Alzer, G. Felder / J. Math. Anal. Appl ) Case Applying Lemmas 1, 3, 5 and log5) 1/5 > 1gives H)>ψ ) ψ) 4 ) 1 + 2ψ) 3 ψ ) > 1 log) 1 ) 4 ] ) 1 + 2log) 3 12 Q ), say. 2.5) Moreover, we have ) 4 log) 4 Q ) log) log) log) + 2 ) 2 ) log5) log5) log5) ) ) 25 From 2.5) and 2.6) we obtain H)>for 5. Lemma 8. For all > we have 1 ) min 1 t) t 2 2.7) Proof. Let Then, 1 )>. Applying Lemma 6 gives for 1.5: This yields 1 ) 1 ) + ) 1 ) 1 ) ) 1 ) min 1 ) Furthermore, we have min 1 ) 1 ) Thus, min ) Net, we show that 1 ).639 for, 1.5] 2, ). If, 1.5], then 1 ) Ɣ) 2 ψ ) Ɣ ) 2 ψ 1.5) Let 2. We define τ ) Ɣ).498 and 2.9. Since τ is conve on 2, ) and τ )>, we get τ ) τ ) + )τ ) τ ) + 2 )τ ).5... Hence, Ɣ) >.498. Using this estimate and Lemma 2ii) we obtain ) 2 Ɣ) 1 ) 2 ψ ) ] >.498) 2 4ψ 2) The proof of Lemma 8 is complete.

6 H. Alzer, G. Felder / J. Math. Anal. Appl ) Main result With the help of Lemmas 7 and 8 we are now in a position to determine the best possible constant lower bound α in inequality 1.1). Theorem. For all odd integers n 1 and real numbers > we have α Ɣ n 1) )Ɣ n+1) ) Ɣ n) ) 2, 3.1) with the best possible constant α min Ɣ) 2 ψ ) ) Proof. We have n ) e t t 1 logt) n 1 dt e t t 1 logt) n+1 dt e s t st) 1 logs) n 1 logt) n+1 logs) n logt) n] dsdt e t t 1 logt) n dt e s t st) 1 logs) logt) ] n 1 logt) 2 logs) logt) ] dsdt 1 e s t st) 1 logs) logt) ] n 1 ] 2 logs) logt) dsdt. 2 Let k 1 be an integer. We get the integral representation 2k+1 ) 2k 1 ) 1 2 e s t st) 1 logs) 2 logt) 2] k 1 logs) 2 logt) 2 1 ] logs) logt) ] 2 dsdt. ) 2 Using the elementary inequality z k 1 z 1) z 1 z, k 1, 2,...) with z logs) 2 logt) 2 gives 2k+1 ) 2k 1 ) 3 ) 1 ). 3.3) From 3.3) and Lemma 7 we obtain 2k 1 )< 2k+1 ) >, k 1, 2,...). 3.4) Combining 3.4) with 2.7) yields 3.1). Moreover, we conclude that the lower bound given in 3.2) is best possible. Remarks. 1) In view of 3.1) it is natural to look for an upper bound for n ), which is valid for all odd n 1 and real >. We show that such a bound does not eist. Using the Leibniz rule for differentiation gives for k : We have Ɣ k) ) 1 Ɣ + 1) ) k) k ) k ) 1 ν) Ɣ k ν) + 1). ν 1 ) ν) 1) ν ν! ν 1, ν so that we obtain k 1 k+1 Ɣ k) k ) 1) ν) ν ν! k ν Ɣ k ν) + 1) + 1) k k!ɣ + 1). ν

7 282 H. Alzer, G. Felder / J. Math. Anal. Appl ) This yields Since lim k+1 Ɣ k) ) 1) k k!. 2n+2 n ) n Ɣ n 1) ) n+2 Ɣ n+1) ) n+1 Ɣ n) ) ] 2, we get for n 1: lim 2n+2 n ) n 1)!n!. This leads to lim n ). 3.5) 2) From 3.5) we also conclude that there is no constant upper bound for n ), which holds for all even integers n 2 and positive real numbers. Does there eist a lower bound, which is independent of n and? Wehave ).9, ) 6, ) 97, ) 2493, ) It is tempting to conjecture that there is no real number c such that we have n ) c for all even n 2 and >. Acknowledgment We thank the referee for helpful comments. References 1] M. Abramowitz, I.A. Stegun Eds.), Handbook of Mathematical Functions with Formulas, Graphs, and Tables, Dover, New York, ] H. Alzer, On some inequalities for the gamma and psi functions, Math. Comp ) ] H. Alzer, A power mean inequality for the gamma function, Monatsh. Math ) ] H. Alzer, Mean-value inequalities for the polygamma functions, Aequationes Math ) ] H. Alzer, J. Wells, Inequalities for the polygamma functions, SIAM J. Math. Anal ) ] Á. Baricz, Turán type inequalities for generalized complete elliptic integrals, Math. Z ) ] G. Csordas, T.S. Norfolk, R.S. Varga, The Riemann hypothesis and the Turán inequalities, Trans. Amer. Math. Soc ) ] D.W. DeTemple, J.M. Robertson, On generalized symmetric means of two variables, Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz ) ] D.K. Dimitrov, Higher order Turán inequalities, Proc. Amer. Math. Soc ) ] C.J. Eliezer, D.E. Daykin, Generalizations and applications of Cauchy Schwarz inequalities, Quart. J. Math. Oford 2) ) ] K. Engel, On the average rank of an element in a filter of the partition lattice, J. Combin. Theory Ser. A ) ] W. Gautschi, The incomplete gamma function since Tricomi, in: Tricomi s Ideas and Contemporary Applied Mathematics, in: Atti Convegni Lincei, vol. 147, Accad. Naz. Lincei, Rome, 1998, pp ] S. Karlin, G. Szegö, On certain determinants whose elements are orthogonal polynomials, J. Anal. Math /1961) ] A. Laforgia, P. Natalini, Inequalities and Turanians for some special functions, in: Difference Equations, Special Functions and Orthogonal Polynomials, Proc. Int. Conf., Munich, 25, World Scientific, 27, pp ] A. Laforgia, P. Natalini, On some Turán-type inequalities, J. Inequal. Appl. 26), Article ID 29828, 6 pp. 16] A. Laforgia, P. Natalini, Turán-type inequalities for some special functions, J. Inequal. Pure Appl. Math. 7 1) 26), Article 32, 3 pp. 17] S. O Shea, A class of power series with no real zeros, Quart. J. Math. Oford 2) ) ] L. Panaitopol, Erdös Turán type inequalities, J. Inequal. Pure Appl. Math. 4 1) 23), Article 23, 5 pp. 19] D.K. Ross, Inequalities and identities for y 2 n y n 1 y n+1, Aequationes Math ) ] J. Sándor, A bibliography on gamma functions: Inequalities and applications, 21] S. Simic, Turán s inequality for Appell polynomials, J. Inequal. Appl. 26), Article ID 9142, 7 pp. 22] P. Turán, On the zeros of the polynomials of Legendre, Časopis Pěst. Mat. Fys ) ] E.M. Wright, A generalisation of Schur s inequality, Math. Gaz ) 217.