Some conditions for a class of functions to be completely monotonic

Abstract

In this article, we present a necessary condition and a necessary and sufficient condition for a class of functions to be completely monotonic.

1 Introduction and main results

Recall [1] that a function f is said to be completely monotonic on

$$\mathbb {R}^{+}:=(0, \infty)$$

if f has derivatives of all orders on $$\mathbb {R}^{+}$$ and for all $$n\in \mathbb {N}_{0}:=\mathbb {N}\cup\{0\}$$

$$(-1)^{n}f^{(n)}(x) \ge0,\quad x \in \mathbb {R}^{+}.$$

Here and throughout the paper, â„• is the set of all positive integers. The set of all completely monotonic functions on $$\mathbb {R}^{+}$$ is denoted by $$CM(\mathbb {R}^{+})$$.

Bernstein [2] proved that a function f on the interval $$\mathbb{R}^{+}$$ is completely monotonic if and only if there exists an increasing function $$\alpha(t)$$ on $$[0,\infty)$$ such that

$$f(x)= \int_{0}^{\infty}e^{-xt}\,d\alpha(t).$$

Also recall [3] that a positive function f is said to be logarithmically completely monotonic on $$\mathbb {R}^{+}$$ if f has derivatives of all orders on $$\mathbb {R}^{+}$$ and for all $$n\in \mathbb {N}$$

$$(-1)^{n}\bigl[\ln f(x)\bigr]^{(n)}\ge0,\quad x \in \mathbb {R}^{+}.$$

The class of all logarithmically completely monotonic functions on $$\mathbb {R}^{+}$$ is denoted by $$LCM(\mathbb {R}^{+})$$.

It was proved [4] that a logarithmically completely monotonic function is also completely monotonic.

There is a rich literature on completely monotonic, logarithmically completely monotonic functions and their applications. For more recent work, see, for example, [5â€“28].

The Euler gamma function is defined and denoted for $$\operatorname {Re}z>0$$ by

$$\Gamma(z):=\int^{\infty}_{0}t^{z-1} e^{-t}\,dt.$$

The logarithmic derivative of $$\Gamma(z)$$, denoted by

$$\psi(z):= \frac{\Gamma'(z)}{\Gamma(z)},$$

is called the psi or digamma function, and the $$\psi^{(k)}$$ for $$k\in \mathbb {N}$$ are called the polygamma functions.

In this article, we give two necessary conditions and a necessary and sufficient condition for a class of functions

$$f_{a,b,c}(x):=(x+a)\ln x-x-\ln\Gamma(x+b)+c,\quad x\in \mathbb {R}^{+},$$
(1)

where $$a,c\in \mathbb {R}$$, $$b \ge0$$ are parameters, to be completely monotonic. The main results are as follows.

Theorem 1

A necessary condition for the function $$f_{a,b,c}(x)$$ to be completely monotonic on the interval $$(0, \infty)$$ is that

\begin{aligned}& b-a=\frac{1}{2}, \end{aligned}
(2)
\begin{aligned}& 0< b\le\frac{1}{2}, \end{aligned}
(3)

and

$$c\ge\ln\sqrt{2\pi}.$$
(4)

Corollary 1

A necessary condition for the function $$f_{a,b,c}(x)$$ to be completely monotonic on the interval $$(0, \infty)$$ is that

$$-\frac{1}{2}< a\le0.$$
(5)

Theorem 2

For

$$b\in \biggl[\frac{1}{2}-\frac{\sqrt{3}}{6},\frac{1}{2} \biggr],$$

a necessary and sufficient condition for the function $$f_{a,b,c}(x)$$ to be completely monotonic on the interval $$(0, \infty)$$ is that

$$b-a=\frac{1}{2}$$
(6)

and

$$c\ge\ln\sqrt{2\pi}.$$
(7)

2 Lemmas

We need the following lemmas to prove our main results.

Let the Î± be real parameters, Î² a non-negative parameter. Define

$$g_{\alpha,\beta}(x):=\frac{x^{x+\beta-\alpha}}{e^{x}\Gamma(x+\beta)}, \quad x\in \mathbb {R}^{+}.$$

Lemma 1

(see [11])

If

$$g_{\alpha,\beta}\in LCM\bigl(\mathbb {R}^{+}\bigr),$$

then either

$$\beta>0 \quad\textit{and}\quad \alpha\ge\max \biggl\{ \beta,\frac{1}{2} \biggr\}$$

or

$$\beta=0 \quad\textit{and}\quad \alpha\ge1.$$

Lemma 2

(see [7])

Let

$$\beta\in \biggl[\frac{1}{2}-\frac{\sqrt{3}}{6},\frac{1}{2} \biggr].$$

If

$$\alpha\ge\frac{1}{2},$$

then

$$g_{\alpha,\beta}\in LCM\bigl(\mathbb {R}^{+}\bigr).$$

3 Proof of the main results

Proof of Theorem 1

If

$$f_{a,b,c}\in CM\bigl(\mathbb {R}^{+}\bigr),$$

then

$$f_{a,b,c}(x)\ge0,\quad x\in \mathbb {R}^{+},$$
(8)

and $$f_{a,b,c}(x)$$ is decreasing on $$\mathbb {R}^{+}$$.

It is well known that (see [29, p.47])

$$\ln\Gamma(x+\beta)= \biggl(x+\beta-\frac{1}{2} \biggr)\ln x-x+\frac{\ln(2\pi)}{2} +O \biggl(\frac{1}{x} \biggr), \quad\mbox{as } x\to \infty.$$
(9)

Hence

$$f_{a,b,c}(x)= \biggl(\frac{1}{2}-b+a \biggr)\ln x-\ln \sqrt{2\pi}+c+O \biggl(\frac{1}{x} \biggr), \quad\mbox{as } x\to\infty.$$
(10)

From (8) and (10), we get

$$\frac{1}{2}-b+a\ge\frac{\ln\sqrt{2\pi}-c+O(1/x)}{\ln x}, \quad\mbox{as } x\to\infty.$$
(11)

Since

$$\frac{\ln\sqrt{2\pi}-c+O(1/x)}{\ln x} \to0, \quad\mbox{as } x\to\infty,$$
(12)

from (11) we have

$$b-a \le\frac{1}{2}.$$
(13)

On the other hand, since $$f_{a,b,c}(x)$$ is decreasing on $$\mathbb {R}^{+}$$, from (10), we obtain

$$\biggl(\frac{1}{2}-b+a \biggr)\ln x-\ln\sqrt{2\pi}+c+O \biggl(\frac{1}{x} \biggr) \le f_{a,b,c}(\tau), \quad\mbox{as } x\to \infty,$$
(14)

where, in (14), Ï„ is a fixed number in $$\mathbb {R}^{+}$$.

Equation (14) is equivalent to

$$\frac{1}{2}-b+a\le\frac{\ln\sqrt{2\pi}+O(1/x)-c+f_{a,b,c}(\tau)}{\ln x}, \quad\mbox{as } x\to\infty.$$
(15)

It is easy to see that

$$\frac{\ln\sqrt{2\pi}+O(1/x)-c+f_{a,b,c}(\tau)}{\ln x} \to 0, \quad\mbox{as } x\to\infty.$$
(16)

Then from (15) we have

$$b-a \ge\frac{1}{2}.$$
(17)

Combining (13) and (17) gives

$$b-a = \frac{1}{2}.$$
(18)

From (8), (10), and (18), we obtain

$$c-\ln\sqrt{2\pi}\ge O \biggl(\frac{1}{x} \biggr), \quad\mbox{as } x\to\infty.$$
(19)

Since

$$O \biggl(\frac{1}{x} \biggr)\to0, \quad\mbox{as } x\to\infty,$$
(20)

from (19) we have

$$c\ge\ln\sqrt{2\pi}.$$
(21)

We note that

$$f_{a,b,c}(x)=\ln g_{b-a,b}(x)+c.$$
(22)

If

$$f_{a,b,c}\in CM\bigl(\mathbb {R}^{+}\bigr),$$

we can verify that

$$g_{b-a,b}\in LCM\bigl(\mathbb {R}^{+}\bigr).$$

By Lemma 1, if

$$b>\frac{1}{2},$$

then

$$b-a\ge b>\frac{1}{2},$$
(23)

$$b=0,$$

by Lemma 1, we get

$$b-a\ge1,$$
(24)

which is another contradiction to (18). So we have proved that

$$0< b\le\frac{1}{2}.$$
(25)

The proof of Theorem 1 is thus completed.â€ƒâ–¡

Proof of Corollary 1

This follows from (2) and (3).

The proof of Corollary 1 is completed.â€ƒâ–¡

Proof of Theorem 2

By Theorem 1, the condition is necessary.

On the other hand, by Lemma 2, we see that

$$g_{b-a,b}\in LCM\bigl(\mathbb {R}^{+}\bigr).$$

Then from (22), we have, for $$n\in \mathbb {N}$$,

$$(-1)^{n} f^{(n)}_{a,b,c}(x)\ge0,\quad x\in \mathbb {R}^{+}.$$
(26)

In particular,

$$f'_{a,b,c}(x)\le0,\quad x\in \mathbb {R}^{+}.$$
(27)

Hence $$f_{a,b,c}(x)$$ is decreasing on $$\mathbb {R}^{+}$$.

By (9),

$$f_{a,b,c}(x)= \biggl(\frac{1}{2}-b+a \biggr)\ln x+c- \ln\sqrt{2\pi}+O \biggl(\frac{1}{x} \biggr),\quad \mbox{as } x\to\infty.$$
(28)

If

$$b-a=\frac{1}{2}$$

and

$$c \ge\ln\sqrt{2\pi},$$

from (28), we obtain

$$\lim_{x \to\infty}f_{a,b,c}(x)=c-\ln\sqrt{2\pi} \ge0.$$
(29)

Therefore

$$f_{a,b,c}(x)\ge\lim_{x \to\infty}f_{a,b,c}(x) \ge0,\quad x\in \mathbb {R}^{+},$$
(30)

which means that (26) is also valid for $$n=0$$. Hence we have proved that

$$f_{a,b,c}\in CM\bigl(\mathbb {R}^{+}\bigr).$$

The proof of Theorem 2 is hence completed.â€ƒâ–¡

References

2. Bernstein, S: Sur les fonctions absolument monotones. Acta Math. 51, 1-66 (1928)

3. Atanassov, RD, Tsoukrovski, UV: Some properties of a class of logarithmically completely monotonic functions. C. R. Acad. Bulgare Sci. 41, 21-23 (1988)

4. Horn, RA: On infinitely divisible matrices, kernels, and functions. Z. Wahrscheinlichkeitstheor. Verw. Geb. 8, 219-230 (1967)

5. Guo, B-N, Qi, F: A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. J. Korean Math. Soc. 48, 655-667 (2011)

6. Guo, S: A class of logarithmically completely monotonic functions and their applications. J. Appl. Math. 2014, 757462 (2014)

7. Guo, S: Logarithmically completely monotonic functions and applications. Appl. Math. Comput. 221, 169-176 (2013)

8. Guo, S: Some properties of completely monotonic sequences and related interpolation. Appl. Math. Comput. 219, 4958-4962 (2013)

9. Guo, S, Laforgia, A, Batir, N, Luo, Q-M: Completely monotonic and related functions: their applications. J. Appl. Math. 2014, 768516 (2014)

10. Guo, S, Qi, F: A class of logarithmically completely monotonic functions associated with the gamma function. J.Â Comput. Appl. Math. 224, 127-132 (2009)

11. Guo, S, Qi, F, Srivastava, HM: A class of logarithmically completely monotonic functions related to the gamma function with applications. Integral Transforms Spec. Funct. 23, 557-566 (2012)

12. Guo, S, Qi, F, Srivastava, HM: Supplements to a class of logarithmically completely monotonic functions associated with the gamma function. Appl. Math. Comput. 197, 768-774 (2008)

13. Guo, S, Qi, F, Srivastava, HM: Necessary and sufficient conditions for two classes of functions to be logarithmically completely monotonic. Integral Transforms Spec. Funct. 18, 819-826 (2007)

14. Guo, S, Srivastava, HM: A certain function class related to the class of logarithmically completely monotonic functions. Math. Comput. Model. 49, 2073-2079 (2009)

15. Guo, S, Srivastava, HM: A class of logarithmically completely monotonic functions. Appl. Math. Lett. 21, 1134-1141 (2008)

16. Guo, S, Srivastava, HM, Batir, N: A certain class of completely monotonic sequences. Adv. Differ. Equ. 2013, 294 (2013)

17. Guo, S, Srivastava, HM, Cheung, WS: Some properties of functions related to certain classes of completely monotonic functions and logarithmically completely monotonic functions. Filomat 28, 821-828 (2014)

18. Krasniqi, VB, Srivastava, HM, Dragomir, SS: Some complete monotonicity properties for the $$(p,q)$$-gamma function. Appl. Math. Comput. 219, 10538-10547 (2013)

19. Mortici, C: Completely monotone functions and the Wallis ratio. Appl. Math. Lett. 25, 717-722 (2012)

20. Qi, F, Luo, Q-M: Bounds for the ratio of two gamma functions - from Wendelâ€™s and related inequalities to logarithmically completely monotonic functions. Banach J. Math. Anal. 6, 132-158 (2012)

21. Qi, F, Luo, Q-M, Guo, B-N: Complete monotonicity of a function involving the divided difference of digamma functions. Sci. China Math. 56, 2315-2325 (2013)

22. Salem, A: An infinite class of completely monotonic functions involving the q-gamma function. J. Math. Anal. Appl. 406, 392-399 (2013)

23. Salem, A: A completely monotonic function involving q-gamma and q-digamma functions. J. Approx. Theory 164, 971-980 (2012)

24. Sevli, H, Batir, N: Complete monotonicity results for some functions involving the gamma and polygamma functions. Math. Comput. Model. 53, 1771-1775 (2011)

25. Shemyakova, E, Khashin, SI, Jeffrey, DJ: A conjecture concerning a completely monotonic function. Comput. Math. Appl. 60, 1360-1363 (2010)

26. Wei, C-F, Guo, B-N: Complete monotonicity of functions connected with the exponential function and derivatives. Abstr. Appl. Anal. 2014, 851213 (2014)

27. Yang, S: Absolutely (completely) monotonic functions and Jordan-type inequalities. Appl. Math. Lett. 25, 571-574 (2012)

28. Srivastava, HM, Guo, S, Qi, F: Some properties of a class of functions related to completely monotonic functions. Comput. Math. Appl. 64, 1649-1654 (2012)

29. ErdÃ©lyi, A (ed.): Higher Transcendental Functions, vol.Â 1. McGraw-Hill, New York (1953)

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The author thanks the editor and the referees for their valuable suggestions to improve the quality of this paper.

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Correspondence to Senlin Guo.

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Guo, S. Some conditions for a class of functions to be completely monotonic. J Inequal Appl 2015, 11 (2015). https://doi.org/10.1186/s13660-014-0534-y

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• DOI: https://doi.org/10.1186/s13660-014-0534-y

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Keywords

• necessary condition
• necessary and sufficient condition
• completely monotonic function
• gamma function