SOME MONOTONICITY PROPERTIES AND INEQUALITIES FOR

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1 Kragujevac Journal of Mathematic Volume 4 08 Page SOME MONOTONICITY PROPERTIES AND INEQUALITIES FOR THE p k-gamma FUNCTION KWARA NANTOMAH FATON MEROVCI AND SULEMAN NASIRU 3 Abtract. In thi paper the author preent ome complete monotonicity propertie and ome inequalitie involving the p k-analogue of the Gamma function. The etablihed reult provide the p k-generalization for ome reult known in the literature.. Introduction In a recent paper [ the author introduced a p k-analogue of the Gamma function defined for p N k > 0 and x R + a p p Γ pk x = t x tk dt 0 pk. = and atifying the baic propertie:. Γ pk x + k = p +!k p+ pk x k xx + kx + k... x + pk pkx Γ pkx Γ pk ak = p + p ka Γ p a a R + Γ pk k =. Key word and phrae. Gamma function p k-analogue completely monotonic function logarithmically completely monotonic function inequality. 00 Mathematic Subject Claification. Primary: 33B5. Secondary: 6A48 33E50. Received: Augut Accepted: March

2 88 K. NANTOMAH F. MEROVCI AND S. NASIRU The p k-analogue of the Digamma function i defined for x > 0 a.3 ψ pk x = d dx ln Γ pkx = k lnpk nk + x = k lnpk 0 e kp+t e kt e xt dt. Alo the p k-analogue of the Polygamma function are defined a.4 ψ m dm pk x = dx ψ m+ m! pkx = m nk + x m+ e = m+ kp+t t m e xt dt 0 e kt where m N and ψ 0 pk x ψ pkx. The function Γ pk x and ψ pk x atify the following commutative diagram. p Γ pk x Γk x k k Γ p x p Γx p ψ pk x ψk x k k ψ p x p ψx. From the identity. the following relation are etablihed:.5.6 ψ pk x + k ψ pk x = x ψ m pk x + k ψm pk x = m m! m m! m N. x m+ m+ It follow eaily from.3 and.4 that for x > 0 i ψ pk x i increaing; ii ψ m pk x i poitive and decreaing if m i odd; iii ψ m pk x i negative and increaing if m i even. Next we recall the following definition and lemma which will be ued in the paper. A function f i aid to be completely monotonic on an interval I if f ha derivative of all order and atifie.7 m f m x 0 for x I m N 0. If the inequality.7 i trict then f i aid to be trictly completely monotonic on I. A poitive function f i aid to be logarithmically completely monotonic on an interval I if f atifie.8 m [ln fx m 0 for x I m N 0.

3 SOME MONOTONICITY PROPERTIES AND INEQUALITIES 89 If the inequality.8 i trict then f i aid to be trictly logarithmically completely monotonic on I. Lemma. [. If h i completely monotonic on 0 then exp h i alo completely monotonic on 0. Lemma. [. Let a i and b i i =... n be real number uch that 0 < a a a n 0 < b b b n and λ i= a i λ i= b i for λ N. If f i a decreaing and convex function on R then n n fb i fa i. i= Lemma.3 [4. Let f x be completely monotonic on 0. Then for 0 the function µx = exp fx + fx + f x + + ηx = exp fx + fx + f x + + f x + i= are logarithmically completely monotonic on 0. In thi paper our goal i to etablih ome complete monotonicity propertie and ome inequalitie involving the p k-analogue of the Gamma function. For additional information on reult of thi nature one could refer to [3 [8 and the related reference therein.. Main Reult We now preent our finding in thi ection. Theorem.. Let p N k > 0 and m N 0. Then the function ψ pkx i trictly completely monotonic on 0. Proof. It follow directly from.4 that which conclude the proof. m ψ pkx m = m ψ m+ pk x = m = m+ m+ m +! nk + x m+ m +! > 0 nk + x m+ Remark.. It follow from Lemma. that exp ψ pk x i alo completely monotonic.

4 90 K. NANTOMAH F. MEROVCI AND S. NASIRU Theorem.. Let p N k > 0 and a 0. Then the function Qx = ψ pk x + a ψ pk x i trictly completely monotonic on 0. In particular Q i decreaing and convex. Proof. By direct computation we obtain m Qx m = [ m ψ m pk x + a ψm pk x [ p = m m+ m! nk + x + a m+ = m+ m! > 0 [ m+ m! nk + x m+ nk + x + a m+ nk + x m+ which etablihe the reult. In particular Q x = ψ pkx + a ψ pkx 0 ince ψ pkx i decreaing. Hence Q i decreaing. Furthermore Q x = ψ pkx + a ψ pkx 0 implying that Q i convex. Remark.. Theorem. generalize the the previou reult [0 Theorem. In the following theorem we prove a generalization of the reult of Mortici [. Theorem.3. Let p N k > 0 and α 0. Then the function T x = ψ pk x + α ψ pk x α x i trictly completely monotonic on 0. Particularly T i decreaing and convex. Proof. Similarly by direct computation we obtain m T x m [ = m ψ m x + α ψm pk x m+ m! pk α x m+ [ p = m m+ m! nk + x + α m+ m! m+ nk + x + m+ αm! m+ x m+ [ = m+ m! >0. nk + x + α m+ nk + x m+ Hence T i trictly completely monotonic on 0. In particular T x = ψ pkx + α ψ pkx + α x = x + x + pα + α + α x = α + x x + pα + α 0 m+ αm! + x m+

5 SOME MONOTONICITY PROPERTIES AND INEQUALITIES 9 a a reult of.6. Thu T i decreaing. Next Hence T i convex. T x = ψ pkx + α ψ pkx α x 3 = x 3 x + pα + α α 3 x 3 α = x 3 x + pα + α 3 0. Remark.3. By letting p and k = in Theorem.3 we obtain the main reult of [. Theorem.4. Let p N k > 0 m N 0 a i and b i i =... n be uch that 0 < a a a n 0 < b b b n and λ i= a i λ i= b i for λ N. Then the function n Γ pk x + a i Hx = Γ pk x + b i i completely monotonic on 0. i= Proof. Let h be defined by hx = n i= [ln Γ pk x + b i ln Γ pk x + a i. Then for m N 0 we have m h x m = m Since x m n i= n [ [ ψ m pk x + b i ψ m pk x + a i = m m m! i= k + x + b i m+ m m! k + x + a i m+ [ n = m+ m! i= k + x + b i m+ k + x + a i m+ i decreaing and convex on R for m N 0 then by Lemma. we obtain [ n k + x + b i 0. m+ k + x + a i m+ i= Thu m h x m 0 for m N 0. Hence h x i completely monotonic on 0. Then by Lemma. exp hx = i completely monotonic on 0. n i= Γ pk x + a i Γ pk x + b i = Hx.

6 9 K. NANTOMAH F. MEROVCI AND S. NASIRU Remark.4. By letting p in Theorem.4 we obtain the reult of [6 Theorem.6. Remark.5. By letting k = in Theorem.4 we obtain the reult of [7 Theorem 3. Remark.6. By letting p and k = in Theorem.4 we obtain the reult of [ Theorem 0. Theorem.5. Let p N k > 0 and a 0. Then the inequality i atified for x [. 0 < ψ pk x + a ψ pk x ap + + ap + Proof. Let Q be defined a in Theorem.. Since Q i decreaing then for x [ we obtain 0 = x lim Qx < Qx Q = ψ pk a + ψ pk which by.5 yield the deired reult. Theorem.6. Let p N and k > 0. Then the inequality. k ln pkx x + ψ pkx k ln pkx hold for x > 0. Proof. It follow from. that ln Γ pk x + k ln Γ pk x = ln pkx x+pk+k. Let gx = ln Γ pk x. Then by the claical mean value theorem the exit a λ x x + k uch that gx + k gx = ln Γ pkx + k ln Γ pk x = ψ pk λ. k k Since ψ pk x i increaing then for λ x x + k we have ψ pk x ψ pk λ ψ pk x + k which implie ψ pk x k ln pkx ψ pk x + k. Then by.5 we obtain ψ pk x k ln pkx ψ pk x + x yielding the reult.. Remark.7. Let p and k = in.. Then we obtain the reult. ln x x ψx ln x for the claical digamma function ψx.

7 SOME MONOTONICITY PROPERTIES AND INEQUALITIES 93 Theorem.7. Let p N and k > 0. Then the inequality.3 k x ψ pkx k x hold for x > 0. + x Proof. Conider the function ψ pk x on the interval x x + k. By the mean value theorem the exit a c x x + k uch that k x = ψ pkx + k ψ pk x = ψ k pkc. Since ψ pkx i decreaing then for c x x + k we have ψ pkx + k ψ pkc ψ pkx which implie Then by.6 we obtain ψ pkx + k k ψ pkx x + k which reult to.3. x ψ pkx. x ψ pkx Remark.8. Let p and k = in.3. Then we obtain the reult.4 for the trigamma function ψ x. x ψ x x + x Remark.9. The right ide of. and the left ide of.4 are however weaker than the reult obtained in [5 Theorem 3. Theorem.8. Let p N k > 0 and 0. Then the function ux = Γ pkx + Γ pk x + exp ψ pk x + wx = Γ pkx + Γ pk x + exp ψ pk x + + ψ pk x + are logarithmically completely monotonic on 0. Proof. Let fx = ln Γ pk x and recall that f x = ψ pkx i completely monotonic on 0 See Theorem.. Then the reult follow directly from Lemma.3.

8 94 K. NANTOMAH F. MEROVCI AND S. NASIRU Theorem.9. Let p N k > 0 and 0. Then the inequality.5 exp ψ pk x + + ψ pk x + Γ pkx + Γ pk x + i atified for x > 0. exp ψ pk x + + Proof. We employ the Hermite-Hadamard inequality which tate that: if fx i convex on [a b then a + b f b fa + fb fx dx. b a a Let fx = ψ pk x a = x + and b = x +. Then we have ψ pk x + + which implie x+ x+ ψ pk x + + ψ pk x + ψ pk t dt ψ pkx + + ψ pk x + ln Γ pkx + Γ pk x + ψ pk Then by taking exponent we obtain the deired reult. x + + Remark.0. By letting p in Theorem.8 and.9 we repectively obtain the reult of Theorem. and.3 of [6. Remark.. By letting k = in Theorem.8 and.9 we repectively obtain the reult of Theorem.3 and.4 of [9. Remark.. The q-analogue of thee reult can alo be found in [4. The following theorem i a p k-generalization of Lemma. of [. We derive our reult by uing imilar technique. Theorem.0. Let p N and k > 0. Then the function fx = [Γ pk x + k x i logarithmically completely monotonic on 0. Proof. We employ the Leibniz rule for n-time differentiable function u and v which i given by n n [uxvx n = u xv n x..

9 SOME MONOTONICITY PROPERTIES AND INEQUALITIES 95 That i [ n ln fx n = ln Γ pk x + k x = = n n = x n+ ln Γ pkx + k n x n n!x n ψ n pk x + k φx. xn+ Thi implie n n φ x =!n x n ψ n pk x + k n n +!x n ψ n pk x + k n = = n!n x n ψ n pk x + k + x n ψ n pk x + k + n n = n!x n ψ n pk x + k n!n x n ψ n n + x n ψ n pk x + k + n n [ n n + x n ψ n pk x + k n + + pk x + k + +!x n ψ n pk x + k + =x n ψ n pk x + k =x n n+ n! k + + x. n+ Suppoe that n i odd. Then!n x n ψ n pk x + k φ x > 0 = φx > φ0 = 0 = ln fx n < 0. Thu n ln fx n > 0. Alo uppoe that n i even. Then φ x < 0 = φx < φ0 = 0 = ln fx n > 0

10 96 K. NANTOMAH F. MEROVCI AND S. NASIRU yielding n ln fx n > 0. Therefore for every n N we have n ln fx n > 0 which conclude the proof. Remark.3. By letting p in Theorem.0 we recover the reult of Theorem.8 of [6. Remark.4. By letting k = in Theorem.0 we recover the reult of Theorem. of [9. 3. Concluion In the tudy the author etablihed ome complete monotonicity propertie and ome inequalitie involving the p k-analogue of the Gamma function which wa recently introduced in [. The etablihed reult provide the p k-generalization for ome reult known in the literature. Reference [ H. Alzer On ome inequalitie for the gamma and pi function Math. Comp [ C-P. Chen X. Li and F. Qi A logarithmically completely monotonic function involving the Gamma function Gen. Math [3 Y.-C. Chen T. Manour and Q. Zou On the complete monotonicity of quotient of Gamma function Math. Inequal. Appl [4 P. Gao Some monotonicity propertie of Gamma and q-gamma function ISRN Math. Anal. 0 0 Article ID page. [5 S. Guo and F. Qi More upplement to a cla of logarithmically completely monotonic function aociated with the Gamma function arxiv: [math.ca. [6 C. G. Kokologiannaki and V. Kraniqi Some propertie of the k-gamma function Matematiche Catania [7 V. Kraniqi T. Manour and A. Sh. Shabani Some monotonicity propertie and inequalitie for Γ and ζ Function Math. Commun [8 V. Kraniqi T. Manour and A. Sh. Shabani Some inequalitie for q-polygamma function and ζ q -Riemann zeta function Ann. Math. Inform [9 V. Kraniqi and F. Merovci Logarithmically completely monotonic function involving the generalized Gamma function Matematiche Catania [0 F. Merovci Sharp inequalitie involving ψ p x function Acta Univ. Apuleni Math. Inform [ C. Mortici A harp inequality involving the Pi function Acta Univ. Apuleni Math. Inform [ K. Nantomah E. Prempeh and S. B. Twum On a p k-analogue of the Gamma function and ome aociated Inequalitie Moroccan J. Pure Appl. Anal

11 SOME MONOTONICITY PROPERTIES AND INEQUALITIES 97 Department of Mathematic Faculty of Mathematical Science Univerity for Development Studie Navrongo Campu P.O. Box 4 Navrongo UE/R Ghana addre: mykwaraoft@yahoo.com knantomah@ud.edu.gh Department of Mathematic Univerity of Mitrovica "Ia Boletini" Koovo addre: fmerovci@yahoo.com 3 Department of Statitic Faculty of Mathematical Science Univerity for Development Studie Navrongo Campu P.O. Box 4 Navrongo UE/R Ghana addre: ulemantat@gmail.com