Complete monotonicity involving some ratios of gamma functions

In this paper, by using the properties of an auxiliary function, we mainly present the necessary and sufficient conditions for various ratios constructed by gamma functions to be respectively completely and logarithmically completely monotonic. As consequences, these not only unify and improve certain known results including Qi’s and Ismail’s conclusions, but also can generate some new inequalities.


Introduction
It is well known that the classical Euler gamma function is defined by for x > , and its logarithmic derivative ψ(x) = (x)/ (x) is known as the psi or digamma function, while ψ , ψ , . . . are called polygamma functions.
Over the past decades, various bounds concerning certain ratios of gamma functions have been researched by many mathematicians. As a possible origin, Wendel [] showed that, for s ∈ (, ) and x > , the following double inequalities hold: Based on a different motivation from Wendel [], Gautschi [] in  independently got the two double inequalities: for n ∈ N and  ≤ s ≤ , one has e (s-)ψ(n+) < (n + s) (n + ) < n s- ,  n +  -s < (n + s) (n + ) <  n -s . In , the above inequalities were improved in [] to be as follows: for  < s <  and x > , More inequalities involving the ratios of above two gamma functions can be found in Qi's review article [] and the references therein. Indeed, these inequalities are almost derived by way of following the monotonicity or convexity properties of the ratios of gamma functions. Ismail et al. [, ] further realized that these inequalities are also the consequences of complete monotonicity of such gamma functions' ratios. Now let us recall that a function f is called completely monotonic (for short, CM) on an interval I if f has the derivative of any order on I and satisfies for all k ≥  on I, see [, ]. An important criterion from the definition is so-called famous Bernstein's theorem, which stated that a necessary and sufficient condition for f (x) to be a completely monotonic for  < x < ∞ is that for nondecreasing α(t), As a classical result, Ismail et al. [] in  showed that the function for any a > b ≥  is logarithmically completely monotonic on (, ∞) if and only if a + b ≥ . Meanwhile, Bustoz and Ismail [] further presented other complete monotonicity results involving the ratio of two gamma functions as follows.
for any a, b ≥  is logarithmically completely monotonic on (, ∞).
More complete monotonicity results concerning the combinations of gamma functions can be found in [-].
Remark . Function (.) can be regarded to be a generalization of Gurland's ratio defined by which appeared in Gurland to give an improvement of Bustoz and Ismail's Theorem A [, Theorem ]. More precisely, they proved the following theorem.
Later, Qi and Guo [, Theorem ] obtained a generalization of Theorem B by establishing the logarithmically complete monotonicity of the following function: for x > -min(s, t, θ (s, t)) with s = t, and they derived the following.
Theorem E ([, Theorem ]) Let s and t be two real numbers with s = t and θ (s, t) be a constant depending on s and t. Then the following statements are valid: Additionally, Qi's another result involving the logarithmically complete monotonicity of the function m s,t can be also found in [, Theorem ]. An improvement of Theorem E and new proofs of Theorems D and E can be found in recent papers [, ].
Inspired by the above mentioned results, we aim to present the necessary and sufficient conditions or sufficient conditions for these ratios W u,v /W r,s , W u,v / n i= W λ i r i ,s i and n i= (W u i ,v i /W r i ,s i ) to be logarithmically completely monotonic on (-min(u, v, r, s), ∞), The rest of this paper is organized as follows. In Section , we introduce an auxiliary function y u,v : and show its properties. These properties, especially Property ., give a necessary and sufficient condition for y u,v (t) ≤ y r,s (t) to hold for all t > , which is crucial to the proof of our main results. In Section , by using Properties . and . of y u,v (t), the necessary and sufficient conditions for ln(W r,s /W u,v ) and sufficient conditions for ln(W u,v / n i= W λ i r i ,s i ) to be completely monotonic are realized respectively, which not only improves certain known results including Qi's and Ismail's conclusions, but also generates some new results. In the fourth section, by means of Properties . and . of y u,v (t) and other two techniques, we further establish the necessary and sufficient conditions for the ratios to be logarithmically completely monotonic.

An important auxiliary function
It is easy to check that y u,v (t) defined by (.) has the following two simple properties.
Then y u,v (t) satisfies the following relations: Now let us further add other two important properties of y u,v (t) which are useful to our main proof.
Proof (i) Using integral representation (.), we immediately get which implies the complete monotonicity of y u,v (t) with respect to u. By the symmetry of u and v, the function y u,v (t) has the same complete monotonicity in parameter v.
(ii) Let φ(x) = -utx-vt(-x). By integral representation (.), ln |y u,v (t)| can be expressed as Likewise, we have These indicate that Remark . Property . together with part (i) of Property . implies that the function The next result, Property ., plays an essential role in proving our main theorems. To prove it, we need the following lemma. Proof Let p = |u -v|/, q = |r -s|/. In the case of (uv)(rs) = , we use the hyperbolic function representation (.) to obtain which is also true for which proves the desired assertion.

Main results
Now we are in a position to state and prove our results. Proof To prove the desired result, we first give the following integral representation: where y u,v (t) is defined by (.). In fact, using the integral representation of ln (x) [, p., (..)] which is obviously valid for u = v, and then (.) follows. The integral representation (.) and Property . indicate that These imply that the inequalities (.) hold, which completes the proof.
While s = r + , the function W u,v /W r,r+ is reduced to By Theorem . we have the following corollary.
Corollary . Let u, v, r ∈ R and ρ = min(u, v, r). Then we have

Remark . Yang and Chu [] showed that ln(W
. It is easy to check that Therefore, these results are equivalent to Qi's Theorem D.
For s = r, the function W u,v /W r,r can be expressed as As a direct consequence of Theorem ., we obtain the following corollary.

An application of Theorem . gives a generalization of Bustoz and Ismail's Theorem C (see [, Lemma ]).
Corollary . Let a, c ∈ R and b ≥ . Then the function if and only if a ≥ c. In particular, if a ≥ c = , then the function On the basis of Property ., we can deduce the following theorem. Proof By the integral representation (.) we have dt.
(ii) While (r i , s i ) = (r, s i ), ρ = ρ  , we have if and only if I(t) ≥  for all t > . The sufficiency obviously follows from the decreasing and convexity of s → y r,s (t) for t > . The necessity can be deduced by the limit relation If s ≤ min ≤i≤n (s i ), then by the decreasing property of s → y r,s (t) we get y r-ρ  ,s i -ρ  (t) ≤ y r-ρ  ,s-ρ  (t) for  ≤ i ≤ n, and then This completes the proof.
Note that . By Theorems . and ., we obtain the following corollary.
Corollary . For fixed u, v, r i , s i ∈ R, i = , , . . . , n, and ρ = min ≤i≤n (u, v, r i , s i ), let the function W u,v be defined on (-min(u, v), ∞) by (.). Then, for λ i >  with n i= λ i = , the function ln

Further results
In [], Grinshpan and Ismail considered the logarithmically complete monotonicity of a more general combination of gamma functions. More precisely, they proved the following theorem. In fact, our main results presented in Section  are essentially regarded as some special cases of the above theorem. Generally speaking, it is very hard to determine those parameters α k , β k . Here we would like to adopt other techniques to determine those parameters such that certain combinations of gamma functions are logarithmically completely monotonic in two special cases.
The first case is that α n = max(α k ) >  and α k ≤  for  ≤ k ≤ n -. In this case, we have (-α k /α n ) ≥  with n- k= (-α k /α n ) = . To this end, we need the following basic fact.
where M t (a, λ) is defined by (.) with a i = e -(s i -ρ) and ρ = min ≤i≤n (s, s i ). By Bernstein's theorem, we see that -[ln g  (x)] is completely monotonic on (-ρ, ∞) if and only if M t (a, λ) te -(s-ρ)t ≥  for all t > , which is equivalent to for all t > . Using Lemma ., the necessary and sufficient condition for -[ln g  (x)] to be completely monotonic on (-ρ, ∞) is that which implies that s ≥ n i= λ i s i . Similarly, the necessary and sufficient condition for [ln g  (x)] to be completely monotonic on (-ρ, ∞) is that which yields that s ≤ min ≤i≤n (s i ), and the proof is complete.
Note that which yields that as x → ∞, In order to ensure that lim x→∞ [ln g  (x)] ≥ , it is necessary toss ≥ . This relation in combination with Theorem . gives the following theorem. The second case is that n is an even number and α k- = , α k = - for k = , , . . . , n/. Theorem C [, Theorem ] is clearly a direct result in this case, and as a generalization of Alzer's work [, Theorem ] which was proved in  as follows.
Theorem G ([, Theorem ]) Let a i and b i be the real numbers such that  ≤ a  ≤ a  ≤ · · · ≤ a n ,  ≤ b  ≤ b  ≤ · · · ≤ b n , and k i= a i ≤ k i= b i for k = , , . . . , n. Then the function is logarithmically completely monotonic on (, ∞).
To prove Theorem G, Alzer made use of the following lemma. . . , n) be real numbers such that a  ≤ a  ≤ · · · ≤ a n , b  ≤ b  ≤ · · · ≤ b n and k i= a i ≤ k i= b i for k = , , . . . , n. If the function f is decreasing and convex on R, then Here we slightly improve Alzer's Theorem G by way of Lemmas . and ..
Theorem . Let u i , r i ∈ R, i = , , . . . , n, such that u  ≤ u  ≤ · · · ≤ u n , r  ≤ r  ≤ · · · ≤ r n , k i= u i ≤ k i= r i for k = , , . . . , n -. Then the function if and only if u  ≤ r  and n i= u i ≤ n i= r i .
Proof It suffices to prove that -[ln g  (x)] ∈ C[(min(u  , r  ), ∞)] if and only if u  ≤ r  and n i= u i ≤ n i= r i . By the integral representation (.) we have where M t (a, λ) is defined by (.) with a i = e -(u i -ρ) , b i = e -(r i -ρ) , λ i = /n and ρ = min ≤i≤n (u i , r i ) = min(u  , r  ). For the necessity, note that -[ln g  (x)] is completely monotonic on (-ρ, ∞), by Bernstein's theorem the inequality or equivalently, holds for all t > . Therefore, by Lemma . we have which are equivalent to n i= u i ≤ n i= r i and u  ≤ r  .
Since u → e -ut is strictly decreasing and convex on R for t > , using Lemma . and Bernstein's theorem, the sufficiency follows. This completes the proof.
If u i ≤ r i (i = , , . . . , n), then k i= u i ≤ k i= r i for k = , , . . . , n. By Theorem . we get the following corollary. Hence, if lim x→∞ ln g  (x) ≥ , then there must be n i= u i -n i= r i ≥ . This together with Theorem . yields the following theorem.
Theorem . Let u i , r i ∈ R, i = , , . . . , n, such that u  ≤ u  ≤ · · · ≤ u n , r  ≤ r  ≤ · · · ≤ r n , Taking n =  in the above two theorems, we conclude the following results.
(i) The function if and only if min(u, v) ≤ min(r, s) and u + v ≤ r + s.
(ii) The function Finally, we give the following theorem by employing the decreasing and convex properties of v → y u,v (t) on R for t >  and Lemma ..