SUPPLEMENTS TO KNOWN MONOTONICITY RESULTS AND INEQUALITIES FOR THE GAMMA AND INCOMPLETE GAMMA FUNCTIONS A. LAFORGIA AND P. NATALINI Received 29 June 25; Acceted 3 July 25 We denote by ΓaandΓa;z the gamma and the incomlete gamma functions, resectively. In this aer we rove some monotonicity results for the gamma function and etend, to >, a lower bound established by Elbert and Laforgia 2 for the function e t dt = [Γ/ Γ/; ]/,with>, only for <<93 +/42 + /. Coyright 26 A. Laforgia and P. Natalini. This is an oen access article distributed under the Creative Commons Attribution License, which ermits unrestricted use, distribution, and reroduction in any medium, rovided the original work is roerly cited.. Introduction and background In a aer of 984, Kershaw and Laforgia [4] investigated, for real α and ositive,some monotonicity roerties of the function α [Γ + /] where, as usual, Γ denotes the gamma function defined by Γa = e t t a dt, a>.. In articular they roved that for >andα = the function [Γ + /] decreases with, while when α= the function [Γ + /] increases. Moreover they also showed that the values α = andα =, in the roerties mentioned above, cannot be imroved if,+. In this aer we continue the investigation on the monotonicity roerties for the gamma functionroving,in Section 2, the following theorem. Theorem.. The functions f = Γ +/, g = [Γ +/] and h = Γ + / decrease for <<, while increase for >. In Section 3, we etend a result reviously established by Elbert and Laforgia [2] related to a lower bound for the integral function e t dt with >. This function can be eressed by the gamma function. and incomlete gamma function defined by Γa;z = e t t a dt, a>, z>..2 z Hindawi Publishing Cororation Journal of Inequalities and Alications Volume 26, Article ID 48727, Pages 8 DOI.55/JIA/26/48727
2 Sulements to the gamma and incomlete gamma functions In fact we have e t dt = Γ/ Γ /;..3 If = 2 it reduces, by means of a multilicative constant, to the well-known error function erf erf = 2 e t2 dt.4 π or to the comlementary error function erf c erf c = 2 e t2 dt = 2 e t2 dt..5 π π Many authors established inequalities for the function e t dt. Gautschi [3] roved the following lower and uer bounds where >, and [ +2 / ] <e e t dt a [ 2 2 + ],.6 a [ a = Γ + ] /..7 Theintegralin.6 can be eressed in the following way Alzer [] foundthefollowinginequalities Γ + where >, >and e t dt = Γ ; = Γ e t dt..8 e / < e t dt < Γ + e α /,.9 [ α = Γ + ].. Feng Qi and Sen-lin Guo [5] establisched, among others, the following lower bounds for > 2 +e e t dt,.
A. Laforgia and P. Natalini 3 if << / /, while / +e / + / e + // /2 e t dt,.2 2 if > / /. Elbert and Laforgia established in [2] the following estimations for the functions e t dt and e t dt + u + < e t dt < + u, for>, >,.3 v + < 93 + / e t dt, for<<, >,.4 42 + where u = e t dt, v = t In Section 3 we rove the following etension of the lower bound.4. Theorem.2. For >, the inequality.4 holdsfor >. e t dt..5 t We conclude this aer, Section 4, showing some numerical results related to this last theorem. 2. Proof of Theorem. Proof. It is easy to note that min > +/ = 2, consequently Γ +/ > forevery >. We have f = Γ 2 +. 2. Since f < for, and f > for> it follows that f decreasesfor <<, while increases for >. Now consider G = log[g]. We have G = log[γ +/]. Then [ G = log Γ + ] + ψ +, G = 2ψ + + ψ 2 + 2.2. Since G = andg > for>itfollowsthatg < for, and G > for,+. Therefore G, and consequently g, decrease for < <, while increase for >. Finally h = Γ 2 +. 2.3
4 Sulements to the gamma and incomlete gamma functions Since Γ +/ >, hence h < for, and h > for>. It follows that hdecreaseson<<, while increases for >. 3. Proof of Theorem.2 By means the series eansion of the eonential function e t,wehave e t dt = v = n n+ n+n!, n= n n n= nn!, 3. consequently the inequality.4 is equivalent to the following that is, + n= n n nn! < n n+ n+n!, 3.2 n= 2 + + +2 2! Since for every integer n 3 + < +3 3! + + 2 2 +2! 3 3 +3! +. 3.3 n+n! n +n! = n +n n!n+, 3.4 by utting z = the inequality.4isequivalentto sz = n n + n+n n! zn > ; 3.5 n=2 it is clear that the series to the right-hand side of 3.5 isconvergentforanyz R. We can observe that, for >, +s 3 z = 3 n n n+n n! zn = z 2 n=2 42 + z 93 + > 3.6 when <z<93 +/42 +. As a consequence of a well known roerty of Leibniz tye series we have <s 3 z <szfor<z<93 +/42 + just like was roved by Elbert and Laforgia in [2]. It is easy to observe that z = reresents a relative minimum oint for the function szdefinedin3.5. In fact we have sz > forz<and<z<93 +/42 +. Now we can rove Theorem.2 by using the following lemma. Lemma 3.. The function sz, defined in3.5, have not any relative maimum oint in the interval,+.
Proof. For any n consider the artial sum of series 3.5 +s 2n z = 2n k=2 and multily this eression by z / ;wehave z / +s 2n z = 2n k=2 Deriving and dividing by z / we obtain + s 2n z+zs 2nz = A new derivation give us the following eression A. Laforgia and P. Natalini 5 k k k+k k! zk 3.7 k k k k!k+/ zk+/. 3.8 2n k=2 + +s 2nz+zs 2nz = k k k k! zk. 3.9 2n k=2 k k z k. 3. k! Dividing by z and re-writing, in equivalent way, the indees into the sum to the righthand side, the last eression yields + + s 2n 2 2nz + s z 2nz = k k + k +2! zk. 3. Now consider the following series k k + k +2! zk ; 3.2 we have for every z R k= k= k= k k + k +2! zk = k z k k +! k z k k +2! k= = z 2 + z2 = z k= 3! z3 4! + 2 z 3! + z2 4! z3 5! + z z2 2 + z3 3! z4 4! + z 2 z 2 2 z3 3! + z4 4! z5 5! + = e z z e z +z z 2 = f z z 2, 3.3 where f z = z +e z.
6 Sulements to the gamma and incomlete gamma functions Since f = and f z = ze z > forz>, it follows that f z > z,+. From 3., by n +,weobtain + + s z + s z = f z z z 2, 3.4 for every z R. If we assume that z> is a relative maimum oint of szthens z = and s z <, but this roduces an evident contradiction when we substitute z = z in 3.4. Proof of Theorem.2. Since sz > z,93 + /42 +, if we assume the eistence of a oint z>93 +/42 +suchthats z < then there eists at least a oint ζ 93 +/42 +, z suchthatsζ =. Let ζ, eventually, be the smallest ositive zero of sz, hence we have s = sζ = andsz > z,ζ. It follows therefore, that there eists a relative maimum oint z,ζ for the function sz, but this is in contradiction whit Lemma 3.. 4. Concluding remark on Theorem.2 In this concluding section we reort some numerical results, obtained by means the comuter algebra system Mathematica, which justify the imortance of the result obtained by means of Theorem.2.Webrieflyut I = e t dt, 4. while denote with A = Γ + e / 4.2 the lower bound established by Alzer [], with G = [ Γ e a 2 + ] a 4.3 that one established by Gautschi [3], with Q = 2 / + e / + / e + // /2 4.4 that one established by Qi-Guo [5]when> / /, and finally with E = v + 4.5 that one established by Elbert-Laforgia [2].
A. Laforgia and P. Natalini 7 Therefore the following numerical results are obtained: i for = 5 and =.26 > 93 +/42 + / =.23456, we have I E =.272222, I A =.47332, I G =.877, 4.6 I Q =.334; ii for = and =.3 > 93 +/42 + / =.222, I E =.69398, I A =.25222, I G =.725, I Q =.38547; 4.7 iii for = 2 and =.65 > 93 +/42 + / =.6, I E =.73853, I A =.73, I G =.644, I Q =.32265. 4.8 In these three numerical eamles we can note that there eist values of >93 + /42 + / such that E reresents the best lower bound of I withresectto A, Q, and G. Moreover we state that this is always true in general, more reciously we state the following conjecture: for any >, there eists a right neighbourhood of 93 +/42 + / such that E reresents the best lower bound of I with resect to A, Q, and G. References [] H. Alzer, On some inequalities for the incomlete gamma function, Mathematics of Comutation 66 997, no. 28, 77 778. [2] Á. Elbert and A. Laforgia, An inequality for the roduct of two integrals relating to the incomlete gamma function, Journal of Inequalities and Alications 5 2, no., 39 5. [3] W. Gautschi, Some elementary inequalities relating to the gamma and incomlete gamma function, Journal of Mathematics and Physics 38 959, 77 8. [4] D. Kershaw and A. Laforgia, Monotonicity results for the gamma function, Atti della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematiche e Naturali 9 985, no. 3-4, 27 33 986.
8 Sulements to the gamma and incomlete gamma functions [5] F. Qi and S.-L. Guo, Inequalities for the incomlete gamma and related functions, Mathematical Inequalities & Alications 2 999, no., 47 53. A. Laforgia: Deartment of Mathematics, Roma Tre University, Largo San Leonardo Murialdo, 46 Rome, Italy E-mail address: laforgia@mat.uniroma3.it P. Natalini: Deartment of Mathematics, Roma Tre University, Largo San Leonardo Murialdo, 46 Rome, Italy E-mail address: natalini@mat.uniroma3.it