ON SOME INEQUALITIES FOR THE INCOMPLETE GAMMA FUNCTION

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1 MATHEMATICS OF COMPUTATION Volume 66, Number 28, April 997, Pages S (97)84-4 ON SOME INEQUALITIES FOR THE INCOMPLETE GAMMA FUNCTION HORST ALZER Abstract. Let p be a positive real number. We determine all real numbers α = α(p) β= β(p) such that the inequalities [ e βp ] /p < e tp dt < [ e αp ] /p Γ( + /p) are valid for all >. And, we determine all real numbers a b such that log( e a e t ) dt log( e b ) t hold for all >.. Introduction In 955, J. T. Chu [] presented sharp upper lower bounds for the error function erf() = 2 π e t2 dt. He proved that the inequalities (.) [ e r2 ] /2 erf() [ e s2 ] /2 are valid for all if only if r s 4/π. The right-h inequality of (.) (with s = 4/π) was proved independently by J. D. Williams (946) G. Pólya (949); see []. An interesting survey on inequalities involving the complementary error function erfc() = 2 π e t2 dt related functions is given in [4, pp. 77 8]. In particular, one can find inequalities for Mills ratio e 2 /2 /2 e t2 dt, derived by several authors. In 959, W. Gautschi [3] provided upper lower bounds for the more general epression (.2) I p () =e p e tp dt. He established that the double-inequality (.3) 2 [(p +2) /p ] <I p () c p [( p +/c p ) /p ] (with c p = [Γ( + /p)] p/(p ) ) holds for all real numbers p>. It has been pointed out in [3] that the integral in (.2) for p = 3 occurs in heat transfer problems, for p = 4 in the study of electrical discharge through gases. We note Received by the editor May, 995, in revised form, April 5, Mathematics Subject Classification. Primary 33B2; Secondary 26D7, 26D5. Key words phrases. Incomplete gamma function, eponential integral, inequalities. 77 c 997 American Mathematical Society

2 772 HORST ALZER that the integral e tp dt can be epressed in terms of the incomplete gamma function namely, Γ(a, ) = t a e t dt, e tp dt = p Γ ( p,p ). Gautschi [3] showed that the inequalities (.3) can be used to derive bounds for the eponential integral E () =Γ(,). Indeed, if p tends to, then (.3) leads to ( 2 log + 2 ) ( e E () log + ) (.4) ( << ). It is the main purpose of this paper to establish new inequalities for e tp dt e tp dt. In Section 2 we present sharp upper lower bounds for Γ( + /p) e tp dt Γ( + /p) e tp dt, which are valid not only for p>, but also for p (, ). In particular, we obtain an etension of Chu s double-inequality (.). Moreover, we provide sharp inequalities for the eponential integral E (). Finally, in Section 3 we compare our bounds with those given in (.3) (.4). 2. Main results First, we generalize the inequalities (.). Theorem. Let p be a positive real number, let α = α(p) β = β(p) be given by α =, β = [Γ( + /p)] p, if <p<, α = [Γ( + /p)] p, β =, if p>. Then we have for all positive real : [ e βp ] /p (2.) < Γ( + /p) Proof. We have to show that the functions G p () = F p () = e tp dt < [ e αp ] /p. e tp dt Γ( + /p)[ e p ] /p e tp dt +Γ(+/p)[ e ap ] /p (a = [Γ( + /p)] p ) are both positive on (, ), if p>, are both negative on (, ), if <p<. First, we determine the sign of F p (). Differentiation yields e p F p()= Γ( + /p)[l(z())] ( p)/p,

3 ON SOME INEQUALITIES FOR THE INCOMPLETE GAMMA FUNCTION 773 where L(z) =(z )/ log(z) z()=e p. Setting f p () =e p F p(), we obtain (2.2) f p()= p d Γ( + /p)(l(z())) 2+/p p d z() d dz L(z) z=z(). Since d dz L(z) = [log(z) +/z]/(log(z))2 > ( <z ) d z() <, d we conclude from (2.2) that f p() <, if p>, f p() >, if <p<. If p>, then we have f p () = Γ( + /p) > lim p() =, which implies that there eists a number > such that f p () > for (, ) f p () < for (, ). Hence, the function F p () is strictly increasing on [, ] strictly decreasing on [, ). Since F p () = lim F p () =,we obtain F p () > for all >. If <p<, then we have f p () = Γ( + /p) < lim p() =. This implies that there eists a number > such that F p () is strictly decreasing on [, ] strictly increasing on [, ). From F p () = lim F p () = we conclude that F p () < for all >. Net, we consider G p (). Differentiation leads to (2.3) e p G p() = +(y()) /a [L(y())] ( p)/p, where L(y) =(y )/ log(y) y()=e ap with a = a(p) = [Γ( + /p)] p. To determine the sign of G p() we need the inequalities ( < ) p (2.4) a(p) p < for <p. 2 The left-h inequality of (2.4) is obviously true. A simple calculation reveals that the second inequality of (2.4) is equivalent to [ ( ) ] + (2.5) ( ) Γ( +) > for <. 2

4 774 HORST ALZER To establish (2.5) we define for >: Then we have d 2 d 2 g() = d d g() =logγ(+) log +. 2 ψ( +) +2 (+) 2 = n=2 ( + n) 2 + dt < ( + t) 2 + =. Thus, g is strictly concave on [, ). Since g() = g() =, we conclude that g is positive on (, ) negative on (, ). This implies (2.5). Let <r</2; we define for y (, ): h r (y) =y r log(y)/(y ). Then we get Since (y ) 2 y r y h r(y)=[(r )y r] log(y)+y =ϕ r (y), say. 2 y 2ϕ r(y)= r [ y 2 y r ], r r r it follows that ϕ r is strictly conve on (, r ) strictly concave on ( r, ). From lim y ϕ r (y) =, ϕ r () = y ϕ 2 r(y) y= = y 2 ϕ r(y) y= =2r <, we conclude that there eists a number y (, ) such that ϕ r is positive on (,y ) negative on (y, ). Thisimpliesthaty h r (y) is strictly increasing on (,y ) strictly decreasing on (y, ). Since lim y h r (y) = lim y h r (y) =,we conclude that there eists a number y (, ) such that h r (y) < fory (,y ) h r (y) > fory (y,). The function y() =e ap is strictly decreasing on [, ). Since y() = lim y() =, there eists a number > such that y <y()< for (, ), <y()<y for (, ). Hence, we have: If <<,thenh r (y()) >,, if <,thenh r (y()) <. We set r =( a(p) ) p p ; then we obtain from (2.3) that h r (y()) = Therefore, if p>, then G p() > for (, ) [ +e ] p/(p ) p G p(). G p() < for (, );

5 ON SOME INEQUALITIES FOR THE INCOMPLETE GAMMA FUNCTION 775, if <p<, then G p() < for (, ) G p() > for (, ). Since G p () = lim G p () =, we conclude that G p () > for (, ), if p>, G p () < for (, ), if <p<. This completes the proof of Theorem. Remark. It is natural to ask whether the double-inequality (2.) can be refined by replacing α by a positive number which is smaller than { ma{, [Γ( + /p)] p, if <p<, } = [Γ( + /p)] p, if p>, or by replacing β by a number which is greater than { min{, [Γ( + /p)] p [Γ( + /p)] p, if <p<, } =, if p>. We show that the answer is no! Let α>beareal number such that the right-h inequality of (2.) holds for all >. This implies that the function F p () = e tp dt Γ( + /p)[ e αp ] /p is negative on (, ). Since F p () =, we obtain F p () = = α /p Γ( + /p), which leads to α [Γ( + /p)] p.ifα (, ), then we conclude from lim ep F p ()= that there eists a number >such that F p () is negative strictly decreasing on [, ). This contradicts lim Fp () =. Thus,wehaveα ma{, [Γ( + /p)] p }. Net, we suppose that β>isareal number such that the first inequality of (2.) is valid for all >. This implies G p () = e tp dt +Γ(+/p)[ e βp ] /p < for all >. Since G p () =, we obtain G p () = = β /p Γ( + /p), which yields β [Γ( + /p)] p.ifβ>, then we get lim ep G p ()=.

6 776 HORST ALZER This implies that there eists a number > such that G p () is negative strictly decreasing on [, ). This contradicts lim Gp () =. Hence, we get β min{, [Γ( + /p)] p }. As an immediate consequence of Theorem, the Remark, the representation e tp dt =Γ(+/p) e tp dt, we obtain the following sharp bounds for the ratio e tp dt/ e tp dt. Corollary. Let p be a positive real number. The inequalities [ e αp ] /p (2.6) < e tp dt < [ e βp ] /p Γ( + /p) are valid for all positive if only if α ma{, [Γ( + /p)] p } β min{, [Γ( + /p)] p }. Now, we provide new upper lower bounds for the eponential integral E () = e t t dt. Theorem 2. The inequalities (2.7) log( e a ) E () log( e b ) are valid for all positive real if only if a e C <b, where C = is Euler s constant. Proof. The function t log( e t )(>) is strictly decreasing on (, ). Therefore, it suffices to prove (2.7) with a = e C b =. Letp>; from (2.6) with α = [Γ( + /p)] p, β =,instead of p,weobtain Γ(/p)[ ( e α ) /p ] < t +/p e t dt < Γ(/p)[ ( e ) /p ]. If p tends to, thenweget log( e a ) E () log( e ) with a = e C. We assume that there eists a real number b> such that E () log( e b ) holds for all >. Since n e E () = ( ) k (k )! k + r n () (>) with k= r n () <n! n (see [2, pp ]), we obtain (2.8) e log( e b ) r (). If we let tend to, then inequality (2.8) implies. Hence, we have b. Using the representation E () = C log() ( ) n n ( >) n! n n=

7 ON SOME INEQUALITIES FOR THE INCOMPLETE GAMMA FUNCTION 777 (see [2, p. 674]), we conclude from the left-h inequality of (2.7) that log e a C ( ) n n n! n. n= If tends to, then we obtain log(/a) C or a e C. The proof of Theorem 2 is complete. 3. Concluding remarks In the final part of this paper we want to compare the bounds for the integrals e tp dt (p >) e t t dt which are given in (.3), (2.6) (.4), (2.7), respectively. First, we consider the bounds for e tp dt. We define with Then we have This implies Let with From S p () =, R p () =Γ(+/p){ [ e αp ] /p } e p 2 [(p +2) /p ] α = [Γ( + /p)] p p>. R p () = Γ( + /p) 2 +/p >, lim R p() = lim R p () > ep R p()=. for all sufficiently small >, R p () < for all sufficiently large. S p () =Γ(+/p){ [ e p ] /p } ce p [( p +/c) /p ] lim c = [Γ( + /p)] p/(p ) p>. S p() = [Γ( + /p)] p/(p ) Γ( + /p) <, lim S p() = lim ep S p()=, we conclude S p () < for all sufficiently small >, S p () > for all sufficiently large. Hence, for small > the bounds for e tp dt (p >) which are given in (2.6) are better than those presented in (.3), whereas for large values of the opposite is true.

8 778 HORST ALZER Net, we compare the bounds for the eponential integral E (). First, we show that for all > the upper bound given in (.4) is better than the upper bound given in (2.7). This means, we have to prove that (3.) e log( + /) < log( e ) for all >. Using the etended Bernoulli inequality ( + z) t +tz (t >; z> ) (see [4, p. 34]), the elementary inequality e t > +t (t ), we obtain for >: ( + ) e e + e e =+ e >+, which leads immediately to (3.). Finally, we compare the lower bounds for E () given in (2.7) (.4). Let T () = e 2 log( + 2/) + log( e a ) with a = e C. Since lim T () =, weobtaint()< for all sufficiently small. And, from lim T () = lim d ea T () =, d we conclude that T () > for all sufficiently large. Thus, for small >the lower bound for E () which is given in (2.7) is better than the bound given in (.4), while for large values of the latter is better. Acknowledgement I thank the referee for helpful comments. References. J. T. Chu, On bounds for the normal integral, Biometrika 42 (955), MR 6:838f 2. G. M. Fichtenholz, Differential- und Integralrechnung, II, Dt. Verlag Wissensch., Berlin, 979. MR 8f:26 3. W. Gautschi, Some elementary inequalities relating to the gamma incomplete gamma function, J. Math. Phys. 38 (959), MR 2: D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York, 97. MR 43:448 Morsbacher Str., 5545 Waldbröl, Germany