INEQUALITIES FOR BETA AND GAMMA FUNCTIONS VIA SOME CLASSICAL AND NEW INTEGRAL INEQUALITIES

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1 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS VIA SOME CLASSICAL AND NEW INTEGRAL INEQUALITIES S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT Abstrct. In this survey pper we present the nturl ppliction of certin integrl ineulities such s, Chebychev s ineulity for synchronous nd synchronous mppings, Holder s ineulity nd Gruss nd Ostrowski s ineulities for the celebrted Euler s Bet nd Gmm functions. Nturl pplictions deling with some dptive udrture formule which cn be deduced from Ostrowski s ineulity re lso pointed out.. Introduction This survey pper is n ttempt to present the nturl ppliction of certin integrl ineulities such s, Chebychev s ineulity for synchronous nd synchronous mppings, Hölder s ineulity nd Grüss nd Ostrowski s ineulities for the celebrted Euler s Bet nd Gmm functions. In the first section, following the well known book on specil functions by Lrry C. Andrews, we present some fundmentl reltions nd identities for Gmm nd Bet functions which will be used freuently in the seuel. The second section is devoted to the pplictions of some clssicl integrl ineulities for the prticulr cses of Bet nd Gmm functions in their integrl representtions. The first subsection of this is devoted to the pplictions of Chebychev s ineulity for synchronous nd synchronous mppings for Bet nd Gmm functions whilst the second subsection is concerned with some functionl properties of these functions which cn be esily derived by the use of Hölder s ineulity. Applictions of Grüss integrl ineulity, which provides more generl pproch thn Chebychev s ineulity, re considered in the lst subsection. The third nd fourth sections re entirely bsed on some very recent results on Ostrowski type ineulities developed by Drgomir et l. in - 6. It is shown tht Ostrowski s type ineulities cn provide generl udrture formule of the Riemnn type for the Bet function. The reminders of the pproimtion re nlyzed nd upper bounded using different techniues developed for generl clsses of rel mppings. Those sections cn be lso seen themselves s new nd powerful tools in Numericl Anlysis nd the interested reder cn use them for other pplictions besides those considered here. For different pproch on Theory of Ineulities for Gmm nd Bet Functions we recommend the ppers 7-7. Dte: Mrch 4, Mthemtics Subject Clssifiction. Primry 6D5, 6D99. Key words nd phrses. Ineulities for Bet nd Gmm functions, Chebychev s ineulity, Holder s ineulity, Gruss ineulity, Ostrowski s ineulity.

2 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT. Gmm nd Bet Functions.. Introduction. In the eighteenth century, L. Euler ) concerned himself with the problem of interpolting between the numbers n! e t t n dt, n,,,... with non-integer vlues of n. This problem led Euler, in 79, to the now fmous Gmm function, generliztion of the fctoril function tht gives mening to! where is ny positive number. The nottion Γ ) is not due to Euler however, but ws introduced in 89 by A. Legendre ), who ws lso responsible for the Dupliction Formul for the Gmm function. Nerly 5 yers fter Euler s discovery of it, the theory concerning the Gmm function ws gretly epnded by mens of the theory of entire functions developed by K. Weierstrss ). The Gmm function hs severl euivlent definitions, most of which re due to Euler. To begin with, we define, p. 5.) n!n Γ ) lim n + ) + )... + n). If is not zero or negtive integer, it cn be shown tht the limit.) eists, p. 5. It is pprent, however, tht Γ ) cnnot be defined t,,,... since the limit becomes infinite for ny of these vlues. By setting in.) we see tht.) Γ ). Other vlues of Γ ) re not so esily obtined, but the substitution of + for in.) leds to the Recurrence Formul, p. 3.3) Γ + ) Γ ). Eution.3) is the bsic functionl reltion for the Gmm function; it is in the form of difference eution. A direct connection between the Gmm function nd fctorils cn be obtined from.) nd.3).4) Γ n + ) n!, n,,,..... Integrl Representtion. The gmm function rrely ppers in the form.) in pplictions. Insted, it most often rises in the evlution of certin integrls; for emple, Euler ws ble to show tht, p. 53.5) Γ ) e t t dt, >. This integrl representtion of Γ ) is the most common wy in which the Gmm function is now defined. Lstly, we note tht.5) is n improper integrl, due to the infinite limit of integrtion nd becuse the fctor t becomes infinite if

3 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 3 t for vlues of in the intervl < <. None the less, the integrl.5) is uniformly convergent for ll b, where < b <. A conseuence of the uniform convergence of the defining integrl for Γ ) is tht we my differentite the function under the integrl sign to obtin, p. 54.6) nd Γ ) e t t log t dt, >.7) Γ ) e t t log t) dt, >. The integrnd in.6) is positive over the entire intervl of integrtion nd thus it follows tht Γ ) >, i.e., Γ is conve on, ). In ddition to.5), there re vriety of other integrl representtions of Γ ), most of which cn be derived from tht one by simple chnges of vrible, p. 57.8) nd Γ ) log u) du, >.9) By setting y.) π Γ ) Γ y) Γ + y) cos θ sin y θ dθ,, y >. in.9) we deduce the specil vlue Γ ) π..3. Other Specil Formule. A formul involving Gmm functions tht is somewht comprble to the double-ngle formule for trigonometric functions is the Legendre Dupliction Formul, p. 58.) Γ ) Γ + ) πγ ), >. An especilly importnt cse of.) occurs when n n,,,...), p. 55.) Γ n + ) n)! π, n,,,.... n n! Although it ws originlly found by Schlömlich in 844, thirty-two yers before Weierstrss fmous work on entire functions, Weierstrss is usully credited with the infinite product definition of the Gmm function.3) Γ ) eγ n + ) e n n where γ is the Euler-Mscheroni constnt defined by

4 4 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT.4) γ lim n n k k log n An importnt identity involving the Gmm function nd sine function cn now be derived by using.3), p. 6. We obtin the identity,.5) Γ ) Γ ) π non-integer). sin π The following properties of the Gmm function lso hold for emple, see, p p. 65):.6).7) Γ ) s e st t dt,, s > ; Γ ) ep t e t) dt, > ;.8).9).).).).3).4).5) Γ ) e t t d + n ) n n! + n), > ; Γ ) log b) t b t dt, >, b > ; Γ ) Γ ) Γ + ) Γ ), > ; e t t ) t log t dt, > ; ) Γ n )n n n )! π, n )! n,,,... ; ) ) Γ + n Γ n ) n π, n,,,... ; Γ 3) π 33 Γ ) Γ + ) Γ + ), 3 3 > ; Γ ) Γ ) Γ ), >..4. Bet Function. A useful function of two vribles is the Bet function, p. 66 where.6) β, y) : t t) y dt, >, y >. The utility of the Bet function is often overshdowed by tht of the Gmm function, prtly perhps becuse it cn be evluted in terms of the Gmm function. However, since it occurs so freuently in prctice, specil designtion for it is widely ccepted. It is obvious tht the Bet mpping hs the symmetry property

5 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 5.7) β, y) β y, ) nd the following connection between the Bet nd Gmm functions holds.8) β, y) Γ ) Γ y), >, y >. Γ + y) The following properties of the Bet mpping lso hold see for emple, p p. 7).9).3).3) β +, y) + β, y + ) β, y),, y > ; β, y + ) y β +, y) y β, ) β, β, y),, y > ; + y ), > ;.3) Γ ) Γ y) Γ z) Γ w) β, y) β + y, z) β + y + z, w),, y, z, w > ; Γ + y + z + w) + p β, p ) pπ ).33) π sec, < p < ;.34) for, y, p >. β, y) t + t y t + ) +y dt p + p) +y t t) y t + p) +y dt 3. Ineulities for the Gmm nd Bet Functions Vi Some Clssicl Results 3.. Ineulities Vi Chebychev s Ineulity. The following result is well known in the literture s Chebychev s integrl ineulity for synchronous synchronous) mppings. Lemm. Let f, g, h : I R R be so tht h ) for I nd h, hfg, hf nd hg re integrble on I. If f, g re synchronous synchronous) on I, i.e., we recll it 3.) f ) f y)) g ) g y)) ) for ll, y I, then we hve the ineulity 3.) h ) d h ) f ) g ) d ) I I I h ) f ) d h ) g ) d. I A simple proof of this result cn be obtined using Korkine s identity 3 3.3) h ) d h ) f ) g ) d h ) f ) d h ) g ) d I I I I h ) h y) f ) f y)) g ) g y)) ddy. I I

6 6 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT The following result holds see lso 4). Theorem. Let m, n, p, 3.4) Then 3.5) nd 3.6) be positive numbers with the property tht p m) n) ). β p, ) β m, n) ) β p, n) β m, ) Γ p + n) Γ + m) ) Γ p + ) Γ m + n). Proof. Define the mppings f, g, h :,, ) given by Then f ) p m, g ) ) n nd h ) m ) n. f ) p m) p m, g ) n ) ) n,, ). As, by 3.3), p m) n) ), then the mppings f nd g re synchronous synchronous) hving the sme opposite) monotonicity on,. Also, h is nonnegtive on,. Writing Chebychev s ineulity for the bove selection of f, g nd h we get, Tht is, ) m ) n d m ) n p m d ) m ) n d m ) n p m ) n d p ) n d which, vi.6), is euivlent to 3.5). Now, using 3.5) nd.8), we cn stte Γ p) Γ ) Γ p + ) Γ m) Γ n) Γ m + n) which is clerly euivlent to 3.6). m ) n ) n d. p ) d ) Γ p) Γ n) Γ p + n) m ) d, Γ m) Γ ) Γ m + ) The following corollry of Theorem my be noted s well: Corollry. For ny p, m > we hve the ineulities 3.7) nd 3.8) β m, p) β p, p) β m, m) Γ p + m) Γ p) Γ m).

7 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 7 Proof. In Theorem set p nd n m. Then nd thus p m) n) p m) β p, p) β m, m) β p, m) β m, p) β p, m) nd the ineulity 3.7) is proved. The ineulity 3.8) follows by 3.7). The following result employing Chebychev s ineulity on n infinite intervl holds 4. Theorem. Let m, p nd k be rel numbers with m, p > nd p > k > m. If 3.9) then we hve 3.) nd 3.) respectively. k p m k) ), Γ p) Γ m) ) Γ p k) Γ m + k) β p, m) ) β p k, m + k) Proof. Consider the mppings f, g, h :, ), ) given by f ) p k m, g ) k, h ) m e. If the condition 3.9) holds, then we cn ssert tht the mppings f nd g re synchronous synchronous) on, ) nd then, by Chebychev s ineulity for I, ), we cn stte i.e., 3.) m e d ) m e d p k m m e d p e d ) p k m k m e d k m e d p k e d k+m e d. Using the integrl representtion.5), 3.) provides the desired result 3.). On the other hnd, since nd β p, m) β p k, m + k) Γ p) Γ m) Γ p + m) Γ p k) Γ m + k) Γ p + m) we cn esily deduce tht 3.) follows from 3.). The following corollry is interesting.

8 8 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT Corollry. Let p > nd R such tht < p. Then 3.3) nd 3.4) Γ p) Γ p ) Γ p + ) β p, p) β p, p + ). Proof. Choose in Theorem, m p nd k. Then nd by 3.) we get k p m k) Γ p) Γ p ) Γ p + ). The second ineulity follows by the reltion.8). Let us now consider the following definition 4. Definition. The positive rel numbers nd b my be clled similrly oppositely) unitry if 3.5) ) b ) ). Theorem 3. Let, b > nd be similrly oppositely) unitry. Then 3.6) nd 3.7) respectively. Γ + b) ) bγ ) Γ b) β, b) ) b Proof. Consider the mppings f, g, h :, ), ) given by f t) t, g t) t b nd h t) te t. If the condition 3.5) holds, then obviously the mppings f nd g re synchronous synchronous) on, ), nd by Chebychev s integrl ineulity we cn stte tht te t dt t +b e t dt ) provided ) b ) ) ; i.e., 3.8) t e t dt Γ ) Γ + b) ) Γ + ) Γ b + ). t b e t dt Using the recursive reltion.3), we hve Γ + ) Γ ), Γ b + ) bγ b) nd Γ ) nd thus 3.8) becomes 3.6). The ineulity 3.7) follows by 3.6) vi.8). The following corollries my be noted s well: Corollry 3. The mpping ln Γ ) is superdditive for >. Proof. If, b, ), then, by 3.6), ln Γ + b) ln + ln b + ln Γ ) + ln Γ b) ln Γ ) + ln Γ b) which is the superdditivity of the desired mpping.

9 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 9 Corollry 4. For every n N, n nd >, we hve the ineulity 3.9) Γ n) n )! n ) Γ ) n. Proof. Using the ineulity 3.6) successively, we cn stte tht Γ ) Γ ) Γ ) Γ 3) Γ ) Γ ) Γ 4) 3 Γ 3) Γ )... Γ n) n ) Γn ) Γ ). By multiplying these ineulities, we rrive t 3.9). Corollry 5. For ny >, we hve Γ ) Γ + ) 3.). π Proof. We refer to the identity.) from which we cn write Γ ) Γ + ) π Γ ), >. Since Γ ) Γ ), we rrive t Γ ) Γ + ) π Γ ) which is the desired ineulity 3.). For given m >, consider the mpping Γ m :, ) R, The following result holds. Γ m ) Γ + m). Γ m) Theorem 4. The mpping Γ m ) is supermultiplictive on, ). Proof. Consider the mppings f t) t nd g t) t y which re monotonic nondecresing on, ) nd h t) : t m e t is non-negtive on, ). Applying Chebychev s ineulity for the synchronous mppings f, g nd the weight function h, we cn write Tht is, t m e t dt which is euivlent to nd the theorem is proved. t +y+m e t dt t +m e t dt Γ m) Γ + y + m) Γ + m) Γ y + m) Γ m + y) Γ m ) Γ m y) t y+m e t dt.

10 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT 3.. Ineulities Vi Hölder s Ineulity. Let I ) R be n intervl in R nd ssume tht f L p I), g L I) p >, p +, i.e., f s) p ds, g s) ds <. I Then 3.) fg L I) nd the following ineulity due to Hölder holds ) ) f s) g s) ds f s) p p ds g s) ds I I I For proof of this clssic fct using Young type ineulity 3.) y p p +,, y, p + ; s well s some relted results, see the book 3. Using Hölder s ineulity we point out some functionl properties of the mppings Gmm, Bet nd Digmm 5. Theorem 5. Let, b with + b nd, y >. Then 3.3) Γ + by) Γ ) Γ y) b, i.e., the mpping Γ is logrithmiclly conve on, ). Proof. We use the following weighted version of Hölder s ineulity ) f s) g s) h s) ds f s) p p 3.4) h s) ds g s) h s) ds I for p >, p + I integrls eist nd re finite. Choose I I. ) nd h is non-negtive on I nd provided ll the other f s) s ), g s) s by ) nd h s) e s, s, ) in 3.4) to get for I, ) nd p, b ) s ) s by ) e s ds which is clerly euivlent to s +by e s ds nd the ineulity 3.3) is proved. ) ) b s y ) e s ds s by ) b e s ds ) ) b s y e s ds s y e s ds Remrk. Consider the mpping g ) : ln Γ ),, ). We hve g ) Γ ) nd g ) Γ ) Γ ) Γ ) Γ ) Γ ) for, ). Using the ineulity.5) we conclude tht g ) for ll, ) which shows tht Γ is logrithmiclly conve on, ). We prove now similr result for the Bet function 5.

11 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS Theorem 6. The mpping β is logrithmiclly conve on, ) s function of two vribles. Proof. Let p, ), m, n), ) nd, b with + b. We hve Define the mppings β p, ) + b m, n) β p + bm, + bn) t p+bm t) +bn dt t p )+bm ) t) )+bn ) dt t p t) t m t) n b dt. f t) t p t), t, ) g t) t m t) n b, t, ) ) nd choose p, b p + + b, p. i.e., Applying Hölder s ineulity for these selections, we get: t p t) t p t) b dt b t p t) dt t m t) n dt β p, ) + b m, n) β p, ) β m, n) b which is the logrithmic conveity of β on, ). Closely ssocited with the derivtive of the Gmm function is the logrithmicderivtive function, or Digmm function defined by, p. 74 Ψ ) d d log Γ ) Γ ),,,,.... Γ ) The function Ψ ) is lso commonly clled the Psi function. Theorem 7. The Digmm function is monotonic nondecresing nd concve on, ). Proof. As Γ is logrithmiclly conve on, ), then the derivtive of ln Γ, which is the Digmm function, is monotonic nondecresing on, ). To prove the concvity of Ψ, we use the following known representtion of Ψ 6, p.. 3.5) Ψ ) t dt γ, > t

12 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT where γ is the Euler-Mscheroni constnt see.4)). Now, let, y > nd, b with + b. Then 3.6) Ψ + by) + γ t +by dt t t )+by ) dt. t As the mpping R, ) is conve for, ), we cn stte tht 3.7) t )+by ) t + bt y for ll t, ) nd, y >. Using 3.7) we cn obtin, by integrting over t, ), t +by t + bt y ) dt dt t t 3.8) t ) + b t y ) t dt t dt + b t Ψ ) + γ + b Ψ y) + γ Ψ ) + bψ y) + γ. Now, by 3.6) nd 3.8) we deduce i.e., the concvity of Ψ. Ψ + by) Ψ ) + bψ y),, y >,, b, + b ; t y dt t 3.3. Ineulities Vi Grüss Ineulity. In 935, G. Grüss estblished n integrl ineulity which gives n estimtion for the integrl of product in terms of the product of integrls 3, p. 96. Lemm. Let f nd g be two functions defined nd integrble on, b. If 3.9) ϕ f ) Φ, γ g ) Γ for ech, b ; where ϕ, Φ, γ nd Γ re given rel constnts, then b f ) g ) d 3.3) f ) d b b b nd the constnt 4 Φ ϕ) Γ γ) 4 is the best possible. g ) d The following ppliction of Grüss ineulity for the Bet mpping holds 7. Theorem 8. Let m, n, p nd be positive numbers. Then 3.3) β m + p +, n + + ) β m +, n + ) β p +, + ) Proof. Consider the mppings p p 4 p + ) p+ m m n n m + n) m+n. l m,n ) : m ) n, l p, ) : p ),,. In order to pply Grüss ineulity, we need to find the minim nd the mim of l,b, b > ).

13 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 3 We hve d d l,b ) ) b b ) b ) b ) b ) b + b). We observe tht the uniue solution of l,b ) in, ) is +b nd s l,b ) > on, ) nd l,b ) < on, ), we conclude tht is point of mimum for l,b in, ). Conseuently nd M,b : sup, m,b : inf l,b ), ) l,b ) : l,b + b b b + b) +b. Now, if we pply Grüss ineulity for the mppings l m,n nd l p,, we get l m,n ) l p, ) d l m,,n ) d l p, ) d which is euivlent to l m+p,n+ ) d 4 M m,n m m,n ) M p, m p, ) m m n n 4 m + n) m+n p p p + ) p+ nd the ineulity 3.3) is obtined. l m,,n ) d l p, ) d Another simpler ineulity tht we cn derive vi Grüss ineulity is the following. Theorem 9. Let p, >. Then we hve the ineulity β p +, + ) 3.3) p + ) + ) 4 or, euivlently, 3.33) m Proof. Consider the mppings Then, obviously { } 3 p p, β p +, + ) 5 + p + p + 4 p + ) + ) 4 p + ) + ). f ) p, g ) ),,, p, >. inf f ) inf g ) ;,, sup f ) sup g ) ;,,

14 4 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT nd f ) d p +, g ) d +. Using Grüss ineulity we get 3.3). Algebric computtions will show tht 3.3) is euivlent to 3.33). Remrk. Tking into ccount tht β p, ) Γp+) Γp)Γ), the ineulity 3.3) is euivlent to Γ p + ) Γ + ) Γ p + + ) p + ) + ) 4, i.e., p + ) Γ p + ) + ) Γ + ) Γ p + + ) p + ) + ) Γ p + + ) 4 nd s p + ) Γ p + ) Γ p + ), + ) Γ + ) Γ + ), we get 3.34) Γ p + + ) Γ p + ) Γ + ) p + ) + ) Γ p + + ). 4 Grüss ineulity hs weighted version s follows. Lemm 3. Let f, g be s in Lemm nd h :, b, ) such tht h ) d >. Then b 3.35) b h ) d f ) g ) h ) d b b h ) d f ) h ) d h ) d g ) h ) d Γ γ) Φ ϕ) 4 The constnt 4 is best. For proof of this fct which is similr to the clssicl one, see the recent pper 8. Using Lemm 3, we cn stte the following proposition generlizing Theorem 8. Proposition. Let m, n, p, > nd r, s >. Then we hve 3.36) β r +, s + ) β m + p + r +, n + + s + ) β m + r +, n + s + ) β p + r +, + s + ) m m n n 4 m + n) m+n p p p + ) p+ β r +, s + ).

15 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 5 The proof follows by the ineulity 3.35) by choosing h ) l r,s ), f ) l m,n ) nd g ) l p, ),, ). Now, pplying the sme ineulity, but for the mppings h ) l r,s ), f ) p nd g ) ),, ), we deduce the following proposition generlizing Theorem 9. Proposition. Let p, > nd r, s >. Then 3.37) β r +, s + ) β p + r +, + s + ) β p + r +, s + ) β r +, + s + ) 4 β r +, s + ). The weighted version of Grüss ineulity llows us to obtin ineulities directly for the Gmm mpping. Theorem. Let α, β, γ >. Then 3.38) 3 α+β+γ+ Γ α + β + γ + ) Γ γ + ) Γ α + γ + ) Γ β + γ + ) α+β+γ+ 4 αα e α ββ e β Γ γ + ). Proof. Consider the mpping f α t) t α e t defined on, ). Then f α t) αt α e t t α e t e t t α α t) which shows tht f α is incresing on, α) nd decresing on, ) nd the mimum vlue is f α α) αα e. α Using 3.35), we cn stte tht f α t) f β t) f γ t) dt f γ t) dt ) m 4 f α t) min f α t) t, t, f α t) f γ t) dt m f β t) min f β t) t, t, f β t) f γ t) dt ) ) f γ t) dt for ll >, which is euivlent to t α+β+γ e 3t dt e γ e t dt t α+γ e t dt t β+γ e t dt α α ) 4 e α ββ e β e γ e t dt for ll >. As the involved integrls re convergent on, ), we get 3.39) t α+β+γ e 3t dt e γ e t dt t α+γ e t dt t β+γ e t dt α α 4 e α ββ e β e γ e dt) t.

16 6 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT Now, using the chnge of vrible u 3t, we get nd, similrly, nd t α+β+γ e 3t dt 3 u ) α+β+γ e u du Γ α + β + γ + ) 3 3α+β+γ+ t α+γ e t dt Γ α + γ + ) α+γ+ t β+γ e t dt Γ β + γ + ) β+γ+ nd then, by 3.39), we deduce the desired ineulity 3.38). 4. Ineulities for the Gmm nd Bet Functions Vi Some New Results 4.. Ineulities Vi Ostrowski s Ineulity for Lipschitzin Mppings. The following theorem contins the integrl ineulity which is known in the literture s Ostrowski s ineulity see for emple, 9, p. 469). Theorem. Let f :, b R be continuous on, b nd differentible on, b), whose derivtive is bounded on, b) nd let f : sup t,b) f t) <. Then f ) b ) f t) dt b +b ) b ) b ) f for ll, b. The constnt 4 is shrp in the sense tht it cnnot be replced by smller one. The following generliztion of 4.) hs been done in. Theorem. Let u :, b R be L-lipschitzin mpping on, b, i.e., u ) u y) L y for ll, y, b. Then we hve the ineulity 4.) u t) dt u ) b ) L b ) for ll, b. The constnt 4 is the best possible b b ) Proof. Using the integrtion by prts formul for the Riemnn-Stieltjes integrl, we hve nd t ) du t) u ) ) t b) du t) u ) b ) u t) dt u t) dt. )

17 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 7 If we dd the bove two eulities, we get 4.3) u ) b ) u t) dt t ) du t) + t b) du t). d is seuence of di- ) Now, ssume tht n : n) < n) <... < n) n < n) n visions with ν n ) s n, where ν n ) : m i {,...,n } nd ξ n) i n) i, n) i+ v : c, d R is L-Lipschitzin on, b, then 4.4) d c p ) dv ) n) i+ n) i. If p : c, d R is Riemnn integrble on c, d nd lim n ν n) i n lim ν n) i L L lim p p ν n) i d c ξ n) i ξ n) i n p p ) d. ξ n) i ) v n) i+ ) v ) n) i+ n) i ) ) ) n) i+ n) i n) i v ) n) i+ Applying the ineulity 4.4) on, nd, b successively, we get 4.5) t ) du t) + ) v n) i+ n) i n) i ) t b) du t) t ) du t) + t b) du t) b L t dt + t b dt L ) + b ) L b +b ) 4 + b ) nd then, by 4.5), vi the identity 4.3), we get the desired ineulity 4.). To prove the shrpness of the constnt 4, ssume tht the ineulity 4.) holds with constnt C >, i.e., b ) +b 4.6) u t) dt u ) b ) L b ) C + b ) for ll, b. Consider the mpping f :, b R, f ) in 4.6). Then + b ) +b C + b ) b ) ),

18 8 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT for ll, b, nd then for, we get b C + ) b ) 4 which implies tht C 4, nd the theorem is completely proved. The best ineulity we cn get from 4.) is the following one. Corollry 6. Let u :, b R be s bove. Then we hve the ineulity: ) + b 4.7) u t) dt u b ) 4 L b ). The previous results re useful in the estimtion of the reminder for generl udrture formul of the Riemnn type for L-lipschitzin mppings s follows: Let I n : < <... < n < n b be division of the intervl, b nd ξ i i, i+ i,,..., n ) seuence of intermedite points for I n. Construct the Riemnn sums n R n f, I n, ξ) f ξ i ) h i i where h i : i+ i i,,..., n ). We now hve the following udrture formul. Theorem 3. Let f :, b R be n L-lipschitzin mpping on, b nd I n, ξ i, i,,..., n ) be s bove. Then we hve the Riemnn udrture formul 4.8) f ) d R n f, I n, ξ) + W n f, I n, ξ) where the reminder stisfies the estimte n n W n f, I n, ξ) L h i + ξ 4 i ) i + i+ 4.9) i i n L i for ll ξ i i,,..., n ) s bove. The constnt 4 is shrp. Proof. Apply Theorem on the intervl i, i+ to get i+ f ) d fξ i )h i L 4 h i + ξ i ) i + i+. i h i Summing over i from to n nd using the generlized tringle ineulity, we get n i+ W n f, I n, ξ) f ) d fξ i )h i i n L i i 4 h i + ξ i ) i + i+.

19 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 9 Now, s for ll ξ i i, i+ ξ i ) i + i+ 4 h i i,,..., n ), the second prt of 4.9) is lso proved. Note tht, the best estimtion we cn obtin from 4.9) is tht one for which ξ i i+i+, obtining the following midpoint formul. Corollry 7. Let f, I n be s bove. Then we hve the midpoint rule. where f ) d M n f, I n ) + S n f, I n ) n ) i + i+ M n f, I n ) f h i i nd the reminder S n f, I n ) stisfies the estimtion. S n f, I n ) n 4 L h i. Remrk 3. If we ssume tht f :, b R is differentible on, b) nd whose derivtive f is bounded on, b), we cn put insted of L the infinity norm f obtining the estimtion due to Drgomir-Wng from. We re ble now to stte nd prove our results for the Bet mpping. Theorem 4. Let p, > nd,. Then we hve the ineulity β p, ) p ) 4.) m {p, } p )p ) p + 4) p ). Proof. Reconsider the mpping l,b :, ) R, l,b ) ) b. For p, >, we get: l p, t) l p, t) p ) p + ) t, t, ). If t ) p, p+, then l p, t) >. Otherwise, if t ) p p+,, then l p, t) <, which shows tht for t p p+, we hve mimum for l p, nd sup l p, t) l p, t ) p )p ) t,) p + ) p+, p, >. i Conseuently l p, t) lp, t) m p ) p + ) t t, p )p ) p + 4) p+ 4 m {p, }

20 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT for ll t,, nd then 4.) l p, t) m {p, } p )p ) p + 4) p+ 4, p, >. Applying now the ineulity 4.) for f ) l p, ),, nd using the bound 4.), we derive the desired ineulity 4.). The best ineulity we cn get from 4.) is the following. Corollry 8. Let p, >. Then we hve the ineulity: β p, ) p+ 4 m {p, } p )p ) 4.) p + 4) p+ 4. The following pproimtion formul for the Bet mpping holds. Theorem 5. Let I n : < <... < n < n be division of the intervl,, ξ i i, i+ i,,..., n ) seuence of intermedite points for I n nd p, >. Then we hve the formul n β p, ) ξ p i ξ i ) h i + T n p, ) i where the reminder T n p, ) stisfies the estimtion T n p, ) m {p, } p )p ) p + 4) p+ 4 n n h i + ξ 4 i ) i + i+ i i m {p, } p )p ) n p + 4) p+ 4 h i. In prticulr, if we choose for the bove ξ i i + i+, i,,..., n ) ; then we get the pproimtion n β p, ) p+ i + i+ ) p i i+ ) + V n p, ), where i V n p, ) 4 m {p, } p )p ) n p + 4) p+ 4 h i. 4.. Some Ineulities Vi Ostrowski s Ineulity for Mppings of Bounded Vrition. The following ineulity for mppings of bounded vrition 5 holds: Theorem 6. Let u :, b R be mpping of bounded vrition on, b. Then for ll, b, we hve u t) dt u ) b ) b ) + + b b 4.3) u) i i

21 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS where b u) denotes the totl vrition of u. The constnt is the best possible. Proof. Using the integrtion by prts formul for Riemnn-Stieltjes integrl, we hve see lso the proof of the Theorem ) tht 4.4) u ) b ) u t) dt t ) du t) + t b) du t) d is seuence of di- ) for ll, b. Now, ssume tht n : c n) < n) <... < n) n < n) n visions with ν n ) s n, where ν n ) : m i {,...,n } nd ξ n) i n) i, n) i+ is of bounded vrition on, b, then 4.5) d c n) i+ n) i. If p : c, d R is continuous on c, d nd v : c, d R p ) dv ) lim n ν n) i n lim ν n) i sup c,d p p p ) sup n sup p ) c,d ξ n) i ξ n) i ) v ) v n v i d v). c n) i+ Applying 4.5), we hve successively t ) du t) ) u) nd nd then t ) du t) + b t b) du t) b ) u) t b) du t) n) i+ n) i+ ) v ) v ) v t ) du t) + b ) u) + b ) u) b m {, b } u) + u) n) i n) i n) i ) ) ) t b) du t) m {, b } b u)

22 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT b + + b b u). Using the identity 4.4), we get the desired ineulity 4.3). Now, ssume tht the ineulity 4.3) holds with constnt C >, i.e., u t) dt u ) b ) C b ) + + b b 4.6) u), for ll, b. Consider the mpping u :, b R given by { { if, b \ +b } u ) if +b in 4.6). Note tht u is of bounded vrition on, b nd b u), u t) dt nd for +b we get by 4.6) tht C which implies C nd the theorem is completely proved. The following corollries hold. Corollry 9. Let u :, b R be L-lipschitzin mpping on, b. Then we hve the ineulity u t) dt u ) b ) b ) + + b 4.7) f b) f ) for ll, b. The cse of Lipschitzin mppings is embodied in the following corollry. Corollry. Let u :, b R be L-lipschitzin mpping on, b. Then we hve the ineulity u t) dt u ) b ) L b ) + + b 4.8) b ), for ll, b. The following prticulr cse cn be more useful in prctice. Corollry. If u :, b R is continuous nd differentible on, b), u is continuous on, b) nd u : u t) dt <, then u t) dt u ) b ) L b ) + + b 4.9) u for ll, b. Remrk 4. The best ineulity we cn obtin from 4.3) is tht one for +b, obtining the ineulity ) + b u t) dt u b ) b 4.) b ) u).

23 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 3 Now, consider the Riemnn sums n R n f, I n, ξ) f ξ i ) h i where I n : < <... < n < n b is division of the intervl, b nd ξ i i, i+ i,,..., n ) is seuence of intermedite points for I n, h i : i+ i i,,..., n ). We hve the following udrture formul. Theorem 7. Let f :, b R be mpping of bounded vrition on, b nd I n, ξ i i,,..., n ) be s bove. Then we hve the Riemnn udrture formul 4.) i f ) d R n f, I n, ξ) + W n f, I n, ξ) where the reminder stisfies the estimte W n f, I n, ξ) sup i,,...,n h i + ξ i i + i+ b 4.) f) ν h) + sup ν h) b f) ; i,,...,n ξ i i + i+ b f) for ll ξ i i,,..., n ) s bove, where ν h) : m i,,...,n {h i }. The constnt is shrp. Proof. Apply Theorem 6 in the intervl i, i+ to get i+ f ) d f ξ i ) h i h i + ξ i i + i+ 4.3) i i+ i f). Summing over i from to n nd using the generlized tringle ineulity, we get n i+ W n f, I n, ξ) f ) d f ξ i ) h i i i n h i + ξ i i + i+ i sup i,,...,n h i + sup i,,...,n h i + ξ i i + i+ i+ f) i n i i+ i f) ξ i i + i+ b f).

24 4 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT The second ineulity follows by the properties if sup ). Now, s ξ i i + i+ h i for ll ξ i i, i+ i,,..., n ), the lst prt of 4.) is lso proved. Note tht the best estimtion we cn get from 4.) is tht one for which ξ i i+ i+ obtining the following midpoint udrture formul. Corollry. Let f, I n be s bove. Then we hve the midpoint rule where f ) d M n f, I n ) + S n f, I n ) n ) i + i+ M n f, I n ) f h i i nd the reminder S n f, I n ) stisfies the estimtion S n f, I n ) b ν h) f). We re ble now to pply the bove results for Euler s Bet function. Theorem 8. Let p, > nd,. Then we hve the ineulity: β p, ) p ) 4.4) m {p, } β p, ) +. Proof. Consider the mpping l p, t) t p ), t,. We hve for p, > tht nd, s for ll t,, then l p, t) l p, t) p p + ) t p p + ) t m {p, } l p, l p, t) p p + ) t dt m {p, } l p, m {p, } β p, ), p, >. Now, pplying Theorem 6 for u t) l p,, we deduce l p, t) dt p ) + l p, )

25 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 5 m {p, } β p, ) + for ll,, nd the theorem is proved. The best ineulity tht we cn get from 4.4) is embodied in the following corollry. Corollry 3. Let p, >. Then we hve the ineulity 4.5) β p, ) p+ m {p, } β p, ). Now, if we pply Theorem 6 for the mpping l p,, we get the following pproimtion of the Bet function in terms of Riemnn sums. Theorem 9. Let I n : < <... < n < n b be division of the intervl, b, ξ i i, i+ i,,..., n ) seuence of intermedite points for I n, nd p, >. Then we hve the formul 4.6) n β p, ) ξ p i ξ i ) h i + T n p, ) i where the reminder T n p, ) stisfies the estimtion T n p, ) m {p, } ν h) + sup i,,...,n m {p, } ν h) β p, ). ξ i i + i+ β p, ) In prticulr, if we choose bove ξ i i+i+ i,,..., n ), then we get the pproimtion where β p, ) n p+ i + i+ ) p i i+ ) + V n p, ) i V n p, ) m {p, } ν h) β p, ) Ineulities Vi Ostrowski s Ineulity for Absolutely Continuous Mppings Whose Derivtives Belong to L p -Spces. The following theorem concerning Ostrowski s ineulity for bsolutely continuous mppings whose derivtives belong to L p -spces holds see lso ). Theorem. Let f :, b R be n bsolutely continuous mpping for which f L p, b, p >. Then f ) 4.7) f t) dt b ) + ) + b + b ) f + ) b b p b ) f + ) p

26 6 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT for ll, b, where 4.8) f p : f t) p dt ) p. Proof. Integrting by prts, we hve nd t ) f t) dt ) f ) t b) f t) dt b ) f ) If we dd the bove two eulities, we get From this we obtin 4.9) where t ) f t) dt + f ) b p, t) : f t) dt f t) dt. t b) f t) dt b ) f ) f t) dt b { t if t, t b if t, b Now, using Hölder s integrl ineulity, we hve f ) 4.3) f t) dt b b b A simple clcultion shows tht p, t) dt p, t) f t) dt p, t) dt t dt + t ) dt + ) p, t) f t) dt, t, ), b. t b dt t b) dt f t) p dt ) p. f t) dt. )+ + b ) + + ) + ) + b + b ) +. + b b

27 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 7 Now, using the ineulity 4.3), we hve f ) f t) dt b ) + ) + b + b ) + f b + b b p + ) ) + + b ) + b b ) f b p nd the first ineulity in 4.7) is proved. Now, for s nd α < β, consider the mpping h : α, β R defined h ) : α) s + β ) s. Observe tht h ) s α) s β ) s ) nd so h ) < on α, α+β nd h ) > on α+β, β. Therefore, we hve ) α + β inf h ) h β α)s α,β s nd Conseuently, we hve sup h ) h α) h β) β α) s. α,β b ) + + ) + b ) +, α, β nd the lst prt of 4.7) is thus proved. by The best ineulity we cn get from 4.7) is embodied in the following corollry. Corollry 4. Under the bove ssumptions for f, we hve ) + b f f t) dt b b ) 4.3) f + ) p. We now consider the ppliction of 4.7) to some numericl udrture rules. Theorem. Let f be s in Theorem. Then for ny prtition I n : < <... < n < n b of, b nd ny intermedite point vector ξ ξ, ξ,..., ξ n ) stisfying ξi i, i+ i,,..., n ), we hve 4.3) f ) d A R f, I n, ξ) + R R f, I n, ξ) where A R denotes the udrture rules of the Riemnn type defined by n A R f, I n, ξ) f ξ i ) h i, h i : i+ i, i

28 8 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT nd the reminder stisfies the estimte 4.33) R R f, I n, ξ) f p + ) f p + ) n i n i where h i : i+ i i,,..., n ). ξ i i ) + + i+ ξ i ) +) h + i ) Proof. Apply Theorem on the intervls i, i+ i,,..., n ) to get f ξ i) h i i+ f t) dt b i ξi ) + ) + i i+ ξ + i h + i+ ) + ) i f t) p p dt h i h i i for ll i {,,..., n }. Summing over i from to n, using the generlized tringle ineulity nd Hölder s discrete ineulity, we get n i+ R R f, I n, ξ) f ξ i) h i f t) dt n + ) i n + ) i n i+ i f p + ) i i ξ i i ) + + i+ ξ i ) + i+ ξ i i ) + + i+ ξ i ) + ) ) p p f t) p p dt n ξ i i ) + + i+ ξ i ) + i i i ) f t) p p dt nd the first ineulity in 4.33) is proved. The second ineulity follows from the fct tht ξ i i ) + + i+ ξ i ) + for ll i {,,..., n }, nd the theorem is thus proved. h + i The best udrture formul we cn get from the bove generl result is tht one for which ξ i : i+ i+, i,,..., n, obtining the following corollry. Corollry 5. Let f nd I n be s in the bove theorem. Then 4.34) f ) d A M f, I n ) + R M f, I n )

29 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 9 where A M is the midpoint udrture rule, i.e., n ) i + i+ A M f, I n ) : f h i i nd the reminder R M stisfies the estimtion: 4.35) R M f, I n ) f p + ) n i h + i ). We re ble to now pply the bove results for Euler s Bet mpping. Theorem. Let s >, p, > s >. Then we hve the ineulity β p, ) p ) 4.36) l+ + ) l+ l m {p, } l + ) l provided s + l. β s p ) +, s ) + ) s Proof. We pply Theorem for the mpping f t) t p t) l p, t), t, to get 4.37) l + ) l where s > nd s + l. However, s in the proof of Theorem 8, nd then l p, s β p, ) l p, ) l+ + ) l+ l l p, s,, l p, t) l p, t) p p + ) t Using 4.37) we deduce 4.36). ) lp, s t) p p + ) s s ds ) t sp ) t) s ) p p + ) s s ds m {p, } β s p ) +, s ) + ) s. We cn stte now the following result concerning the pproimtion of the Bet function in terms of Riemnn sums. Theorem 3. Let s >, p, > s >. If I n : < <... < n < n is division of,, ξ i i, i+ i,,..., n ) seuence of intermedite points for I n, then we hve the formul 4.38) n β p, ) ξ p i ξ i ) h i + T n p, ) i

30 3 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT where the reminder T n p, ) stisfies the estimte T n p, ) m {p, } β s p ) +, s ) + ) l + ) s l n ξ i i ) l+ + i+ ξ i ) l+) l i m {p, } β s p ) +, s ) + ) l + ) s l n i h l+ i ) l where h i : i+ i i,,..., n ) nd s + l. The proof follows by Theorem pplied for the mpping f t) t p t), t,, nd we omit the detils An Ostrowski Type Ineulity for Monotonic Mppings. The following result of the Ostrowski type holds 3. Theorem 4. Let u :, b R be monotonic nondecresing mpping on, b. Then for ll, b, we hve the ineulity u ) 4.39) u t) dt b { + b) u ) + b sgn t ) u t) dt ) u ) u )) + b ) u b) u )) b + +b u b) u )). b } The ineulities in 4.39) re shrp nd the constnt is the best possible. Proof. Using the integrtion by prts formul for Riemnn-Stieltjes integrl 4.4), we hve the identity 4.4) where u ) b p, t) : u t) dt b { t if t, t b if t, b p, t) du t). b is seuence of di- ) Now, ssume tht n : n) < n) <... < n) n < n) n visions with ν n ) s n, where ν n ) : m i {,...,n } nd ξ n) i n) i, n) i+. n) i+ n) i

31 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 3 If p is Riemnn-Stieltjes integrble by rpport of v, nd v is monotonic nondecresing on, b, then n ) ) ) p ) dv ) lim p ξ n) i v n) i+ v n) 4.4) i ν n) i n ) ) ) v p v lim ν n) i p n lim ν n) i p ) dv ). ξ n) i ξ n) i ) v n) i+ n) i+ Using the bove ineulity, we cn stte tht 4.4) p, t) du t) p, t) du t). Now, let observe tht p, t) du t) t du t) + t ) du t) + t b du t) b t) du t) ) v n) i n) i t ) u t) u t) dt b t) u t) b + u t) dt + b) u ) + b) u ) + u t) dt + u t) dt sgn t ) u t) dt. Using the ineulity 4.4) nd the identity 4.4), we get the first prt of 4.39). We know tht sgn t ) u t) dt u t) dt + As u is monotonic nondecresing on, b, we cn stte tht nd nd then u t) dt ) u ) u t) dt b ) u b) u t) dt. sgn t ) u t) dt b ) u b) ) u ). ))

32 3 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT Conseuently, we cn stte tht + b) u ) + sgn t ) u t) dt + b) u ) + b ) u b) ) u ) b ) u b) u )) + ) u ) u )) nd the second prt of 4.39) is proved. Finlly, let us observe tht b ) u b) u )) + ) u ) u )) m {b, } u b) u ) + u ) u ) b + + b u b) u )) nd the ineulity 4.39) is proved. Assume tht 4.39) holds with constnt C > insted of, i.e., u ) 4.43) u t) dt b { + b) u ) + b sgn t ) u t) dt ) u ) u )) + b ) u b) u )) b +b C + u b) u )). b } Consider the mpping u :, b R given by u ) : { if if, b. Putting in 4.43) u u nd, we get u ) u t) dt b { + b) u ) + b sgn t ) u t) dt ) u ) u )) + b ) u b) u )) b +b C + u b) u )) C + b, which proves the shrpness of the first two ineulities nd the fct tht C should not be less thn. The following corollries re interesting. }

33 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 33 Corollry 6. Let u be s bove. Then we hve the midpoint ineulity ) + b u b u t) dt sgn t + b ) 4.44) u t) dt b b u b) u ). Also, the following trpezoid ineulity for monotonic nondecresing mppings holds. Corollry 7. Under the bove ssumption, we hve u b) + u ) 4.45) u t) dt b u b) u ). Proof. Let us choose in Theorem 4, nd b to obtin u ) b u t) dt b b ) u ) + b u t) dt nd u b) b u t) dt b ) u b) + b u t) dt. Summing the bove ineulities, using the tringle ineulity nd dividing by, we get the desired ineulity 4.45). 5. Ineulities of The Ostrowski Type in Probbility Theory nd Applictions for the Bet Function 5.. An ineulity of Ostrowski s Type for Cumultive Distribution Functions nd Applictions for the Bet Function. Let X be rndom vrible tking vlues in the finite intervl, b, with the cumultive distribution function F ) Pr X ). The following result of Ostrowski type holds 4. Theorem 5. Let X nd F be s bove. Then b E X) 5.) Pr X ) b + b) Pr X ) + b b ) Pr X ) + ) Pr X ) b +b + b sgn t ) F t) dt for ll, b. All the ineulities in 5.) re shrp nd the constnt best possible. is the

34 34 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT Proof. We know, by Theorem 4, tht for monotonic nondecresing mpping u :, b R, we hve the ineulity u ) 5.) u t) dt b { } b + b) u ) + sgn t ) u t) dt b ) u ) u )) + b ) u b) u )) b +b + u b) u )) b for ll, b. Apply 5.) for the monotonic nondecresing mpping u ) F ) nd tke into ccount tht F ), F b), to get F ) 5.3) F t) dt b b + b) F ) + ) F ) + b ) F )) b +b +. b sgn t ) F t) dt However, by the integrtion by prts formul for the Riemnn-Stieltjes integrl, we hve nd E X) : tdf t) tf t) b F t) dt bf b) F ) F t) dt b F ) Pr X ). F t) dt, Then, by 5.3), we get the desired ineulity 5.). To prove the shrpness of the ineulities in 5.), we choose the rndom vrible X such tht F :, R { if F ) : if,. We omit the detils. Remrk 5. Tking into ccount the fct tht Pr X ) Pr X )

35 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 35 then, from 5.), we get the euivlent ineulity E X) 5.4) Pr X ) b { + b) Pr X ) + b b ) Pr X ) + ) Pr X ) b +b + b for ll, b. sgn t ) F t) dt } The following prticulr ineulity cn lso be interesting Pr X + b ) b E X) b sgn t + b ) 5.5) F t) dt nd 5.6) Pr X + b ) E X) b The following corollry my be useful in prctice. sgn t + b ) F t) dt. Corollry 8. Under the bove ssumptions, we hve + b E X) Pr X + b ) + b 5.7) b b Proof. From the ineulity 5.), we get But nd + b E X) b Pr + b E X) b + b E X) b nd the ineulity 5.7) is thus proved. X + b ) Remrk 6. ) Let ε, nd ssume tht 5.8) E X) + b + b E X). b b + + b E X) b ) + b E X) b + b E X) b b E X) b + + b ) + + b E X) b + ε) b ), E X) +.

36 36 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT then 5.9) Pr X + b ) ε. Indeed, if 5.8) holds, then by the right-hnd side of 5.9), we get Pr X + b ) + b E X) + b ε ) b ) + ε. b b) Also, if 5.) E X) + b ε b ), then, by the right-hnd side of 5.7), Pr X + b ) + b Tht is, 5.) ε b ) b E X) ε. Pr X + b ) ε ε, ). The following corollry is lso interesting. b Corollry 9. Under the bove ssumptions of Theorem 5, we hve the ineulity b + sgn t ) 5.) F t) dt Pr X ) b for ll, b. Proof. From the eulity 5.), we hve which is euivlent to Tht is, sgn t ) Pr X ) b E X) b + b) Pr X ) + b F t) dt sgn t ) F t) dt b ) Pr X ) + b) Pr X ) b E X) + sgn t ) F t) dt. b ) Pr X ) b E X) + sgn t ) F t) dt.

37 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 37 Since b E X) F t) dt, then from the bove ineulity, we deduce the first prt of 5.). The second prt of 5.) follows by similr rgument from the ineulity Pr X ) b E X) b + b) Pr X ) + b nd we omit the detils. sgn t ) F t) dt Remrk 7. If we put +b in 5.), then we get b + sgn t + b ) 5.3) F t) dt b Pr X + b ) b sgn t + b ) F t) dt. b We re ble now to give some pplictions for Bet rndom vrible. A Bet rndom vrible X with prmeters p, ) hs the probbility density function f ; p, ) : p ) ; < < β p, ) where Ω {p, ) : p, > } nd β p, ) : tp t) dt. Let us compute the epected vlue of X. We hve E X) The following result holds. p ) d β p, ) β p +, ) p β p, ) p +. Theorem 6. Let X be Bet rndom vrible with prmeters p, ) Ω. Then we hve the ineulities Pr X ) p + + nd Pr X ) p p + + for ll, nd, prticulrly, X Pr ) p +

38 38 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT nd X Pr ) p p +. The proof follows by Theorem 5 pplied to the Bet rndom vrible X. 5.. An Ostrowski Type Ineulity For Probbility Density Function f L p, b. The following theorem holds Theorem 7. Let X be rndom vrible with the probbility density function f :, b R R + nd with cumultive distribution function F ) Pr X ). If f L p, b, p >, then we hve the ineulity: b E X) 5.4) Pr X ) b + f p b ) for ll, b, where p +. ) + + b + f p b ) Proof. By Hölder s integrl ineulity we hve y 5.5) F ) F y) f t) dt y dt y f t) p dt for ll, y, b, where p >, p + nd f p : p b b y f p f t) p dt p ) + is the usul p norm on L p, b. The ineulity 5.5) shows in fct tht the mpping F ) is of r H Hölder type, i.e., 5.6) F ) F y) H y r, ), y, b with < H f p nd r, ). Integrting the ineulity 5.5) over y, b, we get successively F ) 5.7) F y) dy b

39 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 39 F ) F y) dy b b f p y dy b f p y) dy + y ) dy b f p + for ll, b. It is well known tht ) + b f p + f p b ) b ) ) + + b ) + ) b E X) b F t) dt + + b b ) + then, by 5.7), we get the first ineulity in 5.4). For the second ineulity, we observe tht ) + ) b + +, ), b b b nd the theorem is completely proved. Remrk 8. The ineulity 5.4) is euivlent to E X) 5.8) Pr X ) b ) + + f p b ) + b b b + f p b ), ), b. ) + Corollry. Under the bove ssumptions, we hve the double ineulity b + f p b )+ E X) + + f p b ) 5.9) +. Proof. We know tht E X) b. Now, choose in 5.4) to get b E X) b + f p b ) i.e., b E X) + f p b )+ which is euivlent to the first ineulity in 5.9).

40 4 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT i.e., Also, choosing b in 5.4), we get b E X) b + f p b ) E X) + f p b ) + which is euivlent to the second ineulity in 5.9). Remrk 9. We know tht by Hölder s integrl ineulity which gives f t) dt b ) f p f p. b ) Now, if we ssume tht f p is not too lrge, i.e., 5.) then we get nd + b f p + b ) + f p b ) + b + f p b )+ which shows tht the ineulity 5.9) is tighter ineulity thn E X) b when 5.) holds. Another euivlent ineulity to 5.9) which cn be more useful in prctice is the following one: Corollry. With the bove ssumptions, we hve the ineulity: E X) + b b ) + f p b ) 5.). Proof. From the ineulity 5.9) we hve: Tht is, b + b b + f p b )+ E X) + b + b + + f p b )+. + f p b )+ E X) + b b + + f p b )+

41 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 4 which is euivlent to E X) + b nd the ineulity 5.) is proved. + f p b )+ b ) b + f p b ) This corollry provides the possibility of finding sufficient condition in terms of f p p > ) for the epecttion E X) to be close to the men vlue +b. Corollry. Let X nd f be s bove nd ε >. If then f p + ε + ) + b ) b ) + E X) + b ε. The proof is similr, nd we omit the detils. The following corollry of Theorem 7 lso holds: Corollry 3. Let X nd f be s bove. Then we hve the ineulity: Pr X + b ) + ) f p b ) + b E X) + b. Proof. If we choose in 5.4) +b, we get Pr X + b ) b E X) b f p b ) + ) which is clerly euivlent to: Pr X + b ) + E X) + b ) b Now, using the tringle ineulity, we get Pr X + b ) + ) f p b ). Pr X + b ) + E X) + b ) E X) + b ) b b Pr X + b ) + E X) + b ) + b b E X) + b

42 4 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT nd the corollry is proved. + ) f p b ) + + b Finlly, the following result lso holds: Corollry 4. With the bove ssumptions, we hve: E X) + b E X) + b f p b ) + + b ) Pr X + b ) + ). The proof is similr nd we omit the detils. A Bet Rndom Vrible X with prmeters s, t) Ω hs the probbility density function where nd We observe tht, for p >, provided i.e., f ; s, t) p f ; s, t) : s ) t ; < < β s, t) β s, t) : β s, t) β s, t) Ω : {s, t) : s, t > } τ s τ) t dτ. τ ps ) τ) pt ) dτ p τ ps )+ τ) pt )+ dτ β s, t) β p s ) +, p t ) + ) p p s ) +, p t ) + > s > p nd t > p. Now, using Theorem 7, we cn stte the following proposition: p

43 INEQUALITIES FOR BETA AND GAMMA FUNCTIONS 43 Proposition 3. Let p > nd X be Bet rndom vrible with the prmeters s, t), s > p, t > p. Then we hve the ineulity: Pr X ) t 5.) s + t + + ) + β p s ) +, p t ) + ) p + β s, t) for ll,. Prticulrly, we hve X Pr ) t s + t β p s ) +, p t ) + ) + ) β s, t) The proof follows by Theorem 7 choosing f ) f ; s, t),, nd tking into ccount tht E X) s s+t. References L.C. Andrews, Specil Functions for Engineers nd Applied Mthemticins, McMilln Publishing Compny, 985. E.D. Rinville, Specil Functions,, New York, Chelse, D.S. Mitrinović, J.E. Pećrić nd A.M. Fink, Clssicl nd New Ineulities in Anlysis, Kluwer Acdemic Publishers, S.S. Drgomir, M. Drgomir nd K. Prnesh, Chebychev s ineulity pplied to Gmm probbility distributions submitted). 5 S.S. Drgomir, M. Drgomir nd K. Prnesh, Some functionl properties of the gmm nd bet functions submitted). 6 H. Lebedev, Specil Functions nd Their Applictions, Romnin), Ed. Tehhic, Buchrest, S.S. Drgomir, M. Drgomir nd K. Prnesh, Applictions of Grüss ineulities on Euler s Bet nd Gmm Functions, submitted. 8 S.S. Drgomir, Some integrl ineulities of Grüss type, submitted. 9 D.S Mitrinović, J.E. Pećrić nd A.M. Fink, Ineulities for Functions nd Their Integrls nd Derivtives, Kluwer Acdemic Publishers, 994 S.S. Drgomir, On the Ostrowski s Integrl ineulity to Lipschitz mppings nd pplictions, submitted. S.S. Drgomir nd S. Wng, Applictions of Ostrowski s ineulity for the estimtion of error bounds for some specil mens nd for some numericl udrture rules, Applied Mthemtics Letters ) 998), 5-9. S.S. Drgomir nd S. Wng, A new ineulity of Ostrowski s type in Lp norm, Indin Journl of Mthemtics, in press) 3 S.S. Drgomir, Ostrowski s ineulity for monotonic mpping nd pplictions., J. KSIAM, submitted. 4 N.S. Brnett nd S.S. Drgomir, An ineulity of Ostrowski s type for cumultive distribution functions, submitted. 5 S.S. Drgomir, On the Ostrowski s ineulity for mppings of bounded vrition, submitted. 6 S.S. Drgomir, N.S. Brnett nd S. Wng, An Ostrowski type ineulity for rndom vrible whose probbility density function belongs to L p, b, p >, submitted. 7 H. Alzer, Some gmm function ineulities, Mth. Comp. 6993), H. Alzer, On some ineulities for the gmm nd psi functions, Mth. Comp ), H. Alzer, On some ineulities for the incomplete gmm functions, Mth. Comp ), J. Bustoz nd M.E.H. Ismil, On gmm function ineulities, Mth. Comp., 47986), p.

44 44 S. S. DRAGOMIR, R. P. AGARWAL, AND N. S. BARNETT J. Dutk, On some gmm functions ineulities, SIAM J. Mth Anl., 6985), T. Erber, The gmm function ineulities of Gurlnd nd Gutschi, Sknd. Akturietidskr., 4396), W. Gutschi, Some elementry ineulities relting to the gmm nd incomplete gmm function, J. Mth. nd Phys., 38959), W. Gutschi, Some men vlue ineulities for the gmm function, SIAM J. Mth. Anl., 5974), P.J. Grbner, R.F. Tichy nd U.T. Zimmermnn, Ineulities for the gmm function with pplictions to permnents, Discrete Mth., 54996), A. Elbert nd A. Lforgi, On some properties of the gmm function, submitted. 7 F. Qi nd S. Guo, Ineulities for the incomplete gmm nd relted functions, Mth Ine. to pper). S.S. Drgomir) School nd Communictions nd Informtics, Victori University of Technology, PO Bo 448, Melbourne City, MC 8, Victori, Austrli E-mil ddress, S.S. Drgomir: sever@mtild.vu.edu.u URL: Current ddress, R.P. Agrwl: Deprtment of Mthemtics, Ntionl University of Singpore, Kent Ridge Crescent, Singpore, 96. E-mil ddress, R.P. Agrwl: mtrvip@leonis.nus.sg N.S. Brnett) School nd Communictions nd Informtics, Victori University of Technology, PO Bo 448, Melbourne City, MC 8, Victori, Austrli E-mil ddress, N.S. Brnett: neil@mtild.vu.edu.u