Inequalities and Monotonicity For The Ratio of Γ p Functions

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1 Int. J. Open Problems Compt. Math., Vol. 3, No., March 200 ISSN ; Copyright c ICSRS Publication, Inequalities and Monotonicity For The Ratio of Γ p Functions Valmir Krasniqi and Faton Merovci University of Prishtina,Prishtinë 0000, Republic of Kosova vali.99@hotmail.com University of Prishtina,Prishtinë 0000, Republic of Kosova fmerovci@yahoo.com Abstract Let > 0, y 0 be real numbers. The function f() = Γ p(+y+)/γ p(y+) +y+ is strictly decreasing and strictly logarithmically conve on (0, ). Moreover lim f() = eψp(y+) 0 Γ p(+y+)/γ p(y+) Γ p(+y+2)/γ p(y+) + Keywords: Γ p function, Inequalities.. y+ and +y+ +y+2 < Introduction The Euler gamma function Γ() is defined for > 0 by Γ() = 0 t e t dt. Euler gave another equivalent definition for the Γ() (see 5) Γ p () = where lim p Γ p () = Γ(). p!p ( + )... ( + p) = p ( + )... ( + p ) ()

2 2 Valmir Krasniqi and Faton Merovci V. Krasniqi and A. Shabani (see 5) defined the function ψ p () = Γ p() Γ p (). (2) The function ψ p, defined by () has the following series representation ψ p () = ln p p k=0 + k (3) and deriving n times the relation (3) one finds that: ψ (n) p () = p k=0 ( ) n n!. (4) ( + k) n+ In 7, H. Minc and L. Sathre proved that, if r is a positive integer and φ(r) = (r!) r, then φ(r + ) < < r +, (5) φ(r) r which can be rearranged as Γ( + r) r < Γ(2 + r) r+ (6) and Γ( + r) r Γ(2 + r) r+ >. (7) r r + In, 6, H. Alzer and J. S. Martins refined the right inequality in (5) and showed that, if n is a positive integer, then for all positive real numbers r, we have n n + < n n+ n i r i= n+ i r i= r < n n! n+ (n + )!. (8) Both bounds in (8) are the best possible. The inequalities in (5) were refined and generalized in 8, 2, 9, and the following inequalities were obtained: ( n+k n + k + n + m + k + < i=k+ ) ( n / n+m+k i i i=k+ ) n+m n + k n + m + k where k is a nonnegative integer, n and m are natural numbers. For n = m =, the equality in (9) is valid. (9)

3 Inequalities and Monotonicity For the Ratio of Γ p Functions 3 In 8, inequalities in (9) we generalized, where Feng Qi obtained the following inequalities on the ration for the geometric means of a positive arithmetic sequence with unit difference for any nonnegative integer k and natural numbers n and m: n + k + + α n + m + k + + α < ( n+k i=k+ ( n+m+k i=k+ ) n (i + α) (i + α) ) n+m n + k + α n + m + k + α (0) where α 0, is a constant. For n = m =, the equality in (0) is valid. Furthermore, for nonnegative integer k and natural numbers n and m, we have a(n + k + ) + b a(n + m + k + ) + b < ( n+k i=k+ ( n+m+k i=k+ ) n (ai + b) (ai + b) ) n+m a(n + k) + b a(n + m + k) + b, () where a is a positive constant and b a nonnegative integer. For n = m =, the equality in () is valid, (see 3). It is clear that inequalities in () etend those in (0). In 4, the following monotonicity results for the Gamma function were established. The function Γ( + ) decreases with > 0 and Γ( + ) increases with > 0, which recovers the inequalities in (5) which refers to integer value of r. These are equivalent to the function Γ( + ) being increasing and Γ(+) being decreasing on (0, ), respectively. In addition, it was proved that the function γ Γ( + ) decreases for 0 < <, where γ = denotes the Euler s constant, which is equivalent to Γ(+) γ being increasing on (, ). In 2, the following monotonicity result was obtained. The function Γ( + y + )/Γ(y + ) + y + (2) is decreasing for, for fied y 0. Then, for positive real numbers and y, we have + y + + y + 2 Γ( + y + )/Γ(y + ). (3) Γ( + y + 2)/Γ(y + ) +

4 4 Valmir Krasniqi and Faton Merovci 2 Main results The following Theorem is the main result of these notes. Theorem 2. Let > 0, y 0 be real numbers. The function f() = Γ p( + y + )/Γ p (y + ) + y + (4) is strictly decreasing on (0, ). Moreover eψp(y+) lim f() = 0 y + (5) and + y + + y + 2 < Γ p ( + y + )/Γ p (y + ) Γ p ( + y + 2)/Γ p (y + ) +. Taking logarithm yields ln f() = ln Γ p ( + y + ) ln Γ p (y + ) ln( + y + ). For > 0, define h() = 2 f () f() = ln Γ p( + y + ) 2 + ψ p ( + y + ) Γ p (y + ) + y +.

5 Inequalities and Monotonicity For the Ratio of Γ p Functions 5 Differentiation of h gives. h () = ψ p( + y + ) + y + y + ( + y + ) 2 = p+ n= p+ = < n= n= n= n= ( + y + n) 2 + y + y + ( + y + ) 2 ( + y + n) 2 n= y + ( + y + n) y + 2 ( + y + n + ) 2 ( + y + n) 2 n= y + ( + y + n) 2 y + ( + y + n + ) 2 + y + n + y + n + + y + n + y + n + y = ( + y + n) + 2 ( + y + n)( + y + n + ) y + ( + y + n + ) 2 n= (2y + )( + y + n) + y = ( + y + n) 2 ( + y + n + ) < 0. 2 n= Hence, the function h is strictly decreasing and h() < h(0) = 0, for > 0, which yields the desired results that f () < 0, hence f is strictly decreasing on (0, ) and one has the following inequality + y + + y + 2 < Γ p ( + y + )/Γ p (y + ) Γ p ( + y + 2)/Γ p (y + ) Net, by L. Hospital rule, we conclude that +. lim 0 eψp(y+) f() =, y 0. (6) y + Definition 2.2 A function f() is logarithmically conve on the interval a, b if f > 0 and ln f() is conve on a, b Theorem 2.3 The function f given by (4) is strictly logarithmically conve on (0, ).

6 6 Valmir Krasniqi and Faton Merovci For > 0 define g() = 3 d2 ln f() = 2 ln Γ p( + y + ) 2ψ d 2 p ( + y + )+ Γ p (y + ) ψ p( + y + ) + ( + y + ). 2 Differentiation of g yields 2 g () = ψ p ( + y + ) + = n= 2(y + ) + ( + y + ) 2 ( + y + ) 3 p+ 2 = ( + y + n) + 3 ( + y + n) 2 ( + y + n + ) 2 n= n= 2(y + ) + ( + y + n) 2(y + ) 3 ( + y + n + ) 3 n= 2 > ( + y + n) + 3 ( + y + n) 2 ( + y + n + ) 2 + n= n= 2(y + ) ( + y + n) 2(y + ). 3 ( + y + n + ) 3 n= 3(2y + )( + y + n) 2 + (6y + )( + y + n) + 2y ( + y + n) 3 ( + y + n + ) 3 > 0. Hence the function g is strictly increasing and g() > g(0) = 0, for > 0, which yields the desired results, that is d2 (ln f()) d 2 > 0 for > 0. Corollary 2.4 Let y 0 be a real number. Then for all real numbers > 0 Since f is decreasing and / Γ p ( + y + ) Γ p (y + ) + y + eψp(y+) y +. (7) eψp(y+) lim f() = 0 y + (8) we obtain and proof is completed. f() lim 0 f() = eψp(y+) y +

7 Inequalities and Monotonicity For the Ratio of Γ p Functions 7 3 Open Problem At the end, we pose a problem. For positive real numbers and y, holds Γ p ( + y + )/Γ p (y + ) Γ p ( + y + 2)/Γ p (y + ) + + y < + y +. (9) References H. ALZER, On a inequality of H.Minc and L.Sathre, J. Math. Anal. Appl., 79 (993), B. -N. GUO AND F. QI, Inequalities and monotonicity of the ratio of gamma functions, Taiwanese J. Math., 7(2)(2003), B.-N. GUO AND F. QI, Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with arbitrary difference, Tamkang. J. Math., 34(3)(2003), D. KERSHAW AND A. LAFORGIA, Monotonicity results for the gamma function, Atti Accad. SCI. Torino Cl.Sci.Fis.Mat.Natur., 9(985), V. KRASNIQI and A. Sh. SHABANI Some conveity properties and inequalities for the Γ p ()-function, Applied Mathematics Electronic Notes, to appear 6 J. S. MARTINS, Arithmetic and geometric means, and applications to Lorentz sequence spaces, Maths Nachr., 39 (988), H. MINC AND L. SATHRE, Some inequalities involving (r!) r, Proc. Edinburgh Math. Soc., 4 (2) (964/65), F. QI, Inequalities and monotonicity of the ratio for the geometric means of a positive arithmetic sequence with unit differnce, Internat. J. Math. Edu. Sci. Tech.,34, 4, (2003), F. QI AND B.-N. GUO,Some inequalities involving the geometric mean of natural function and sequence RGMIA Res.Rep. Coll., 4()(200), Art. 6, F. QI AND S.-L. GUO, Inequalities for the incomplete gamma and related functions, Math. Inequal. Appl., 2()(999),

8 8 Valmir Krasniqi and Faton Merovci F. QI AND Q.-M. LUO, Generalations of H.Minc and J. Sathre s inequality, Tamkang J. Math., 3(2)(2000),