MATHEMATICS OF COMPUTATION Volume 66, Number 28, April 997, Pages 77 778 S 25-578(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, 996. 99 Mathematics Subject Classification. Primary 33B2; Secondary 26D7, 26D5. Key words phrases. Incomplete gamma function, eponential integral, inequalities. 77 c 997 American Mathematical Society
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,
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
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 (, );
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 ()=.
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 =.5772... 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. 673 674]), 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=
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.
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), 263 265. 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), 77 8. MR 2:267 4. D. S. Mitrinović, Analytic inequalities, Springer-Verlag, New York, 97. MR 43:448 Morsbacher Str., 5545 Waldbröl, Germany