Conditions for convexity of a derivative and some applications to the Gamma function

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1 Aequationes Math. 55 (1998) /98/ $ /0 c Birkhäuser Verlag, Basel, 1998 Aequationes Mathematicae Conditions for conveity of a derivative and some applications to the Gamma function Milan Merkle Summary. We consider necessary and sufficient conditions for the conveity of a function f () in terms of some properties of the associated function of two variables F (, y) = (f(y) f())/(y ). In particular, we prove that f is conve if and only if F is conve and if and only if F is Schur-conve. These results are applied to the theory of the Gamma function. We complement a characterization of the Gamma function due to H. Kairies and present some inequalities for the ratio of Gamma functions. Mathematics Subject Classification (1991). Primary 6A51, 33B15; Secondary 6D10. Keywords. Conveity, Schur-conveity, inequalities, Gamma function. 1. Necessary and sufficient conditions for conveity Let f be a differentiable function defined on an interval I. In this section we investigate a relationship between conveity (or concavity) of f on I and conveity (or concavity) of the divided differences F (, y) = f(y) f() y ( y), F(, ) =f () (1) on I. We shall also consider Schur-conveity of F. For convenience, recall that a symmetric function (, y) g(, y) is Schur-conve on I if and only if g(, y) g( ε, y + ε) for every, y I, ε>0 such that ε, y + ε I. A function g is Schur-concave if g is Schur-conve. For details on Schur-conveity see [11]. Let us consider the following statements: (A) f is conve on I. ( ) + y (B) f F (, y) for all, y I,

2 74 M. Merkle AEM (C) F (, y) f ()+f (y) for all, y I, (D) F is conve on I, (E) F is Schur-conve on I, and (A ) f is concave on I, ( ) + y (B ) f F (, y) for all, y I, (C ) F (, y) f ()+f (y) for all, y I, (D ) F is concave on I, (E ) F is Schur-concave on I. Our main result is the following. Theorem 1. If f () is continuous on I then the conditions (A) (E) are equivalent and the conditions (A ) (E ) are equivalent. For the proof of Theorem 1 we use the lemma below, which presents a slightly modified result from [11, 3.A]. Lemma 1. Let (, y) g(, y) be a symmetric and continuous function on I. Suppose that both partial derivatives eist and are continuous on the set {(, y) I y}. Thengis Schur-conve on I if and only if g(, y) y g(, y) 0 for (, y) I such that <y. () Proof. Since g is continuous, it is Schur-conve if and only if for every fied (, y) in the interior of I, <y, the function ε g( ε, y + ε) is increasing in ε>0. After evaluation the derivative with respect to ε, one finds that the latter condition is equivalent to (). Proof of Theorem 1. (A) (B). By [15, p. 15], a continuous function g is conve on I if and only if g(y) 1 u+h g(t) dt h u h for all u I and h>0 such that u ± h I. Letting here u h =, u + h = y and g = f, we get the desired assertion. (A) (C). By [14, p. 39] or [15, p. 15], a continuous function g is conve on I if and only if 1 y g(t)dt g()+g(y) y

3 Vol. 55 (1998) Conditions for conveity of a derivative 75 for all, y I. Letting here g = f,wegettheassertiontobeproved. (A) (D). Trivially, by f () =F(, ), we have that (D)= (A). So, we need (A)= (D) only. Now let f be continuous and conve on I. Since F is a continuous function in two variables, to show its conveity it suffices to prove that [15, p. 15] F (, y) 1 (F ( ε, y η)+f(+ε, y + η)) (3) holds for any, y I and ε, η > 0sothat±ε, y ± η I. To this end, observe that where f(y) f() = y f(y η) f( ε) = y η+ε y f(y + η) f( + ε) = y +η ε y f (t)dt, (4) y y f (a 1 t+b 1 )dt, (5) f (a t+b )dt, (6) a 1 = y η + ε, a = y y +η ε η yε, b 1 = y y, b = yε η y. Now, since (a 1 t + b 1 + a t + b )/ =tand f is conve, we have f (t) 1 (f (a 1 t + b 1 )+f (a t+b )). By integration over [, y] and using (4) (6), we obtain (3), as desired. (E) (C). By Lemma 1, (E) is equivalent to F F y for <y,,y I and it is easy to see that the latter condition is equivalent to (C). The equivalence of conditions (A ) (E ) follows from the above upon replacing f by f. Theorem 1 can serve as a tool for producing many interesting inequalities. We shall consider some applications related to the Gamma function.

4 76 M. Merkle AEM. A characterization of Gamma function The digamma function Ψ, defined by Ψ() =(logγ()) can be epressed in terms of the series [1] Ψ() = γ n=1 n(n + ) ( >0), where γ is the Euler constant. It follows that Ψ is concave on >0 and therefore, conditions (A ) (E ) hold with f =logγoni=(0,+ ). In this and the net section we investigate some consequences of this fact. In[7]itisprovedthatifgis concave on (0, + ), g( +1) g()=1/ and g(1) = γ then g() Ψ() for>0. In the light of Theorem 1, the requirement for concavity of g =(logγ) can be replaced by either of conditions (B ) (E ) with f = log Γ. Hence, in the net theorem we get a complement to the characterization in [7]. Theorem. Suppose that f is a continuously differentiable real function defined on (0, + ). If one of the conditions (A ) (E ) is satisfied for f on (0, + ) and f( +1) f()=log (>0), then f() =logγ()+c,where Cis an arbitrary real constant. Proof. By Theorem 1, it suffices to assume that f is concave. Then the statement can be proved using the mentioned result of [7], or by a result in [9]. We give an independent simple proof that relies on Bohr Mollerup theorem [3]. Firstly, let us show that f is a non-decreasing function. Indeed, suppose that f (a) >f (b)forsomea<b,a, b > 0. By the assumption, f (b +1) = f (b)+1/b > f (b) and so we have that f (b) =min{f (a),f (b),f (b+1)},which is a contradiction to the concavity of f. Therefore, f is non-decreasing and consequently, f is a conve function. Let h() =e f() f(1). Then h is positive and logarithmically conve on (0, + ), h( +1)=h() andh(1) = 1. By the Bohr Mollerup theorem, h() Γ() and the assertion follows. Note that omitting conditions (B ) (E ) from the statement of Theorem gives the result of Kairies [7]. 3. Some inequalities for a ratio of the Gamma functions Inequalities for the ratio Q(, β) =Γ(+β)/Γ() with >0 and usually β (0, 1), have been studied by many authors (see [, 1] and references therein). Bounds for

5 Vol. 55 (1998) Conditions for conveity of a derivative 77 the ratio Q that involve the function Ψ and its derivatives have been investigated in [, 4, 5, 6, 8, 13], using a variety of methods. In this section we present some new inequalities of this type, starting from (A ) (E ) with f =logγ From (B )and(c ) it follows that 1 log Γ(y) log Γ() (Ψ()+Ψ(y)) Ψ y ( + y ). (7) Letting y = + β, β>0, we get ( ep β Ψ()+Ψ(+β) ) Q(, β) ep(βψ( + β/)). (8) The upper bound in (8) was also obtained in [8] by other means. In [13] we showed that the lower bound in (8) is closer than the lower bound in [8]. 3.. Since (D ) yields (, +1+β)=(1 β)(, +1)+β(, +), F (, +1+β) (1 β)f (, +1)+βF(, +) >0,β [0, 1]. After an application of the recurrence relation Γ(z +1)=zΓ(z) weget Q(, β) (1+β)( β)/ ( +1) β(1+β)/. (9) + β Note the equality in (9) for β =0andβ=1. Weshallshowthatforβ (0, 1), the inequality (9) is sharper than Gautschi s inequality [5] Let us define a function ϕ by (1 + β)( β) ϕ() = log + Then lim ϕ() =0and + Q(, β) ( 1+β) β. (10) β(1 + β) log( +1) log( + β) β log( 1+β). ϕ ()= β(1 β)( + +( β)(1 + β)) ( +1)( 1+β)( + β) < 0 for >1 βand β (0, 1). Therefore, we conclude that ϕ() > 0for>1 βand β (0, 1), which means that (9) is sharper than (10).

6 78 M. Merkle AEM 3.3. Further, from (, + β) =(1 β)(, )+β(, +1) and applying (D )weobtain Q(, β) β ep(β(1 β)ψ()), > 0,β [0, 1]. (11) Using concavity of Ψ and inequality (14) below, it can be proved that this bound is closer than the lower bound in (8) In a similar way, starting from and applying (D ), we get ( + β, + β) =(1 β)( + β,)+β(+β, +1) ( ) Q(, β) β /(1 β) β(1 β) ep Ψ( + β), > 0,β<1/. (1) 1 β 3.5. Condition (E ) implies log Γ(y) log Γ() y log Γ(y + ε) log Γ( ε) y +ε for 0 <<yand 0 <ε<. In particular, replacing by + β and letting y = +β and ε = β, weobtain Γ( +3β) Γ() ( ) Γ( +β), > 0,β>0. (13) Γ( + β) 3.6. Let us now derive some bounds for the function Ψ. Letting y = +1in (7), we get Ψ()+ 1 ( log Ψ + 1 ), whereof it follows ( log 1 ) Ψ() log 1, > 0. (14)

7 Vol. 55 (1998) Conditions for conveity of a derivative An open question The statements (A) (E) presented in Section 1 can be reformulated as (A) F (, ) isconveoni. ( + y (B) F, + y ) F (, y) for all, y I, F (, )+F(y, y) (C) F (, y) for all, y I, (D) F is conve on I, (E) F is Schur-conve on I. where F is defined by (1). It would be interesting to investigate whether or not there eist symmetric functions F that can not be represented in the form (1) and such that all (or some) statements (A) (E) are equivalent. Ecept (D)= (A) and (D)= (E), no other implication holds generally. Note that in the case considered in the present paper, F (, y) is completely determined by its values on the diagonal: F (, y) = 1 y y F(t, t)dt. Acknowledgements. I wish to thank Prof. Ivan Lacković, who provided some references related to this work, including his unpublished monograph [10]. I am also grateful to anonymous referees for their helpful comments and suggestions that led to improvements of the first version of the paper. This research was supported by Science Fund of Serbia, grant number 04M03E, through Mathematical Institute, Belgrade. References [1] M. Abramowitz and I. A. Stegun, A Handbook of Mathematical Functions, NewYork, [] H. Alzer, Some gamma function inequalities, Math. Comp. 60 (1993), [3] Emil Artin, The Gamma Function, Holt, Rinehart and Winston, New York, 1964, translation from the German original of [4] J. Bustoz and M. E. H. Ismail, On gamma function inequalities, Math. Comp. 47 (1986), [5] W. Gautschi, Some elementary inequalities relating to the gamma and incomplete gamma function, J. Math. andphys. 38 (1959), [6] M. E. H. Ismail, L. Lorch and M. E. Muldoon, Completely monotonic functions associated with the Gamma function and its q-analogues, J. Math. Anal. Appl. 116 (1986), 1 9. [7] H. H. Kairies, Über die logarithmische Ableitung der Gammafunktion, Math. Ann. 184 (1970), [8] D. Kershaw, Some etensions of W. Gautschi s inequalities for the gamma function, Math. Comp. 41 (1983),

8 80 M. Merkle AEM [9] W. Krull, Bemerkungen zur Differenzengleichung g(+1) g()=ϕ(), Math. Nachr. 1 (1948), [10] I. B. Lacković, Conve Functions, unpublished manuscript. [11] A. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, Academic Press, New York, [1] M. Merkle, Logarithmic conveity and inequalities for the gamma function, J. Math. Anal. Appl. 03 (1996), [13] M. Merkle, Conveity, Schur-conveity and bounds for the Gamma function involving the Digamma function, Rocky Mountain J. Math., to appear. [14] T. Popoviciu, Les fonctions convees, Actualités Sci. Indust. 99, Paris, [15] A. W. Roberts and D. E. Varberg, Conve functions, Academic Press, New York London, M. Merkle University of Belgrade Faculty of Electrical Engineering Department of Applied Mathematics P. O. Bo Belgrade Yugoslavia Manuscript received: September 1, 1996 and, in final form, February 0, 1997.