Turan inequalities for the digamma and polygamma functions

Transcription

1 South Asia Joural of Mathematics, Vol. : ISSN 5-5 RESEARCH ARTICLE Tura ieualities for the digamma ad polygamma fuctios W.T. SULAIMAN Departmet of Computer Egieerig, College of Egieerig Uiversity of Mosul, Ira. waadsulaima@hotmail.com Received: 8-3-; Accepted: 9-- *Correspodig author Abstract I this ote, may ew ieualities cocerig the digamma ad polygamma fuctios are preseted. Key Words digamma fuctio, polygamma fuctio MSC 6D5 Itroductio The Euler gamma fuctio Γx is defied for x > by Γx The digamma fuctio deoted by Ψx is defied by where Γ deotes the Euler Gamma fuctio. This fuctio has the followig series represetatio where γ is the Euler costat. t x e t.. Ψx Γ x, x >,. Γx Ψx γ + x k k + x + k,. For,,, we deote by Ψ x Ψ x the polygamma fuctio defied as the -th derivative of the fuctio Ψx. This fuctio has the followig itegral represetatio Ψ x + t e t e xt, x >,,, Citatio: W.T. Sulaima, Tura ieualities for the digamma ad polygamma fuctios, South Asia J Math,,,

2 W.T. Sulaima: Tura ieualities for the digamma ad polygamma fuctios Laforgia ad Natalii [4] proved the followig result Theorem.. For x >,,,, the followig ieuality holds where m+ is a iteger. I this paper, may ew ieualities cocerig the digamma ad polygamma fuctios are preseted. Ψ m xψ x Ψ m+ x,.4 Results First we discuss the digamma fuctio. The followig is a geeralizatio of Theorem.. Theorem.. For x >,,,, p +, p >, the followig ieuality holds Ψ /p m xψ / x Ψ m p + x,. where m p + Proof. is a iteger. Ψ m p + x m p + m p t m p + m p + + e t e xt t m p t xt xt e p e e t /p e t / Ψ /p m xψ/x. t m e te xt /p t e t e xt / Theorem. follows from theorem. by puttig p. Theorem.. For x >, < y <, the followig ieuality holds Ψx + y Ψx + Ψy.. Proof. Let fx Ψx + y Ψx Ψy. O keepig y fixed, we have f x Ψ x + y Ψ x x + y + k x + k. k 5

3 South Asia J. Math. Vol. No. Therefore f is o-icreasig. Sice the fx, which implies lim fx lim γ + x x lim x lim x k γ + k + γ + k k x + y k + x + y + k x x + k y y + k k + x + y + k x + k y y + k x + y x y, y + k Next we cosider the polygamma fuctio. Ψx + y Ψx + Ψy. Theorem.3. For x, y >, is a odd iteger, we have If is eve, the ieuality reverses. Ψ xψ y Ψ x + y..3 Proof. It is ot difficult to show that!, is odd, Ψ x k x + k +!.4 x + k +, is eve. Therefore, we have for odd Ψ x! k k x + k +! x + y + k + Ψ x + y..4 Also, Ψ y Ψ x + y. Multiplyig the above ieualities we get k Ψ xψ y Ψ x + y. If is eve, we have Ψ x Ψ x + y Ψ y Ψ x + y Hece, if we multiply the above ieualities we obtai Ψ xψ y Ψ x + y. 5

4 W.T. Sulaima: Tura ieualities for the digamma ad polygamma fuctios Theorem.4. For x, y >, is a odd iteger, the ieuality holds If is eve, the ieuality reverses. Proof. We have Ψ x + Ψ y Ψ x + y..5 Ψ x + Ψ y Ψ x + y! x + k + + y + k + x + y + k + k :! f x. O keepig y fixed, we have f x + x + y + k + x + k +. k Therefore f x is o-icreasig. As lim x f x, the f x. The result follows. Theorem.5. For x, y >, is a odd iteger, p >, p +, we have Ψ x p + y Ψ/p xψ/ y..6 I particular Ψ x + y Ψ xψ y..7 Proof. We have Ψ x p + y t e e t x p +y t p e xt p t e t /p Ψ /p t e xt e t t e xt e t xψ/y. /p / t e yt e t / The particular result follows by puttig p. Theorem.6. For x >, < y <, is a odd iteger, < p <, p + ieuality holds, the followig Ψ xy Ψ /p xψ / y..8 5

5 South Asia J. Math. Vol. No. Proof. First we have to show that x + y xy, x >, < y <..9 For this let gx x + y xy. Let y fixed. The g x y >. That meas g is o-decreasig. Sice g y >, the gx, ad hece.9 proved. Now, for odd, we have Ψ x therefore Ψ x is o-icreasig, which implies Ψ xy Ψ x + y t + e t e xt, t e t e x+yt e yt t p e xt t e t /p e t /p t e xpt e t Ψ /p pxψ / y. Theorem.7. Let x, y >, x + y, p >, p + ieuality holds I particular Ψ x p p + y x + y Ψ If is eve, the above ieualities reverses. Proof. The hypothesis implies that x + y xy. / t e yt e t /, is a odd iteger, the the followig Ψ /p pxψ / y.. Ψ /p x Ψ/ y. x p Ψ p + y t e e t t e t e xyt x p p +y t t e t e x+yt. e yt t p e xt t e t /p e t / 53

6 W.T. Sulaima: Tura ieualities for the digamma ad polygamma fuctios Ψ /p t e t e pxt pxψ/ y. /p t e t e yt / Theorem.8. Let x, y >, x + y, p >, p + ieuality holds Ψ xy Ψ /p, is a odd iteger, the the followig pxψ/ y..3., Proof. By the hypothesis x + y xy. Sice, Ψ x is o-icreasig for odd, the, via steps after Ψ xy Ψ x + y Ψ /p pxψ/y. t e te x+yt Theorem.9. Let x >, < y <, < p <, p +, is a odd iteger, the the followig ieuality holds Ψ xy Ψ /p pxψ / y..4 Proof. As before, the hypothesis implies that x + y xy. As Ψ x is o-icreasig, Ψ xy Ψ x + y t p e xt t e t /p e t t t e te x+yt e yt e te pxt Ψ /p pxψ / y. I the followig, we discuss the Beta fuctio. / /p t e te yt Theorem.. Let a + bx, c + dy >. The the fuctio Ba + bx, c + dy is o-icreasig i both x ad y. 54 Proof. Let fx Ba + bx, c + dy. The lfx l Γ a + bxγc + dy Γ a + c + bx + dy /

7 South Asia J. Math. Vol. No. l Γ a + bx + l Γ c + dy l Γ a + c + bx + dy. O keepig y fixed, differetiatig with respect to x gives f x fx b Γ a + bx Γ a + bx a + c + bx + dy bγ Γ a + c + bx + dy bψ a + bx bψ a + c + bx + dy a + bx a + c + bx + dy b k + a + bx + k a + c + bx + dy + k k k bc + dy. a + bx + ka + c + bx + dy + k As fx >, the f x, ad therefore Ba + bx, c + dy is o-icreasig. Refereces Alsia A. ad Tomas M.S., A geometrical proof of a ew ieuality for the gamma fuctio, J. Ie. Pure Appl. Math., 6 5, Art. 48. Askey R., The -gamma ad -beta fuctios, Applicable Aal., 8, 978/79, Bougoffa L., Some ieualities ivolvig the gamma fuctio, J. Ie. Pure Appl. Math., 7 5 6, Art Laforgia A. ad Natalii P., Tura-type ieualities for some special fuctios, J. Ie. Pure Appl. Math., 7 6, Art Sador J., A ote o certai ieualities for the gamma fuctios, J. Ie. Pure Appl. Math., 6 3 5, Art Sh.Shabai A., Some ieualities for the gamma fuctio, J. Ie. Pure Appl. Math., 8 7, Art