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| Computer Modeling in Engineering & Sciences | |
DOI: 10.32604/cmes.2022.016927
ARTICLE
Complete Monotonicity of Functions Related to Trigamma and Tetragamma Functions
Mona Anis1, Hanan Almuashi2 and Mansour Mahmoud3,*
1Faculty of Science, Mathematics Department, Mansoura University, Mansoura, 35516, Egypt
2Faculty of Science, Mathematics Department, Jeddah University, Jeddah, 21589, Saudi Arabia
3Faculty of Science, Mathematics Department, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
*Corresponding Author: Mansour Mahmoud. Email: mansour@mans.edu.eg
Received: 11 April 2021; Accepted: 20 October 2021
Abstract: In this paper, we study the completely monotonic property of two functions involving the function Δ(x)=[ψ′(x)]2+ψ″(x) and deduce the double inequality x2+3x+33x4(2x+1)2<Δ(x)<625x2+2275x+50433x4(50x+41)2,x>0 which improve some recent results, where ψ(x) is the logarithmic derivative of the Gamma function. Also, we deduce the completely monotonic degree of a function involving ψ′(x).
Keywords: Trigamma function; tetragamma function; completely monotonic function; completely monotonic degree; inequality
1 Introduction
The Euler’s gamma function is defined [1] by the improper integral
Γ(x)=∫0∞e−vvx−1dv,x>0 and the psi or digamma function is defined by the logarithmic derivative of gamma function, that is
ψ(x)=Γ′(x)Γ(x). The two derivatives ψ′(x) and ψ″(x) are respectively called the trigamma and tetragamma functions.
Kirchhoff was the first to apply the polygamma functions ψ(n)(x) in the field of physics [2] and in Feynman calculations arised several series containing polygamma functions [3]. Recently, Wilkins et al. [4] use the digamma function, as well as new variants of the digamma function, as a new family of basis functions in mesh-free numerical methods for solving partial differential equations. The polygamma functions and their inequalities have many interesting applications in physics, statistics and applied mathematics by giving estimates and approximations to the values of some special functions [5].
A function f(x) is said to be completely monotonic [6] (Chapter XIII) on an interval J if the derivatives f(m)(x) exist on J for all m≥0 and
(−1)mf(m)(x)≥0,x∈J,m≥0. According to the Bernstein–Widder theorem [7] the function f(x) is completely monotonic on x≥0 if and only if there is a function μ(v) satisfying that
f(x)=∫0∞e−xvdμ(v), where the integral converges for x≥0 and μ(v) is non-decreasing and bounded, that is, f(x) is completely monotonic on x≥0 if and only if it is a Laplace transform of a non-decreasing and bounded measure μ(v).
In 2004, Alzer [8] presented the inequality
Δ(x)>α(x)900x4(x+1)10,x>0 (1)
where
Δ(x)=[ψ′(x)]2+ψ″(x) and
α(x)=450+3600x+13290x2+29700x3+44101x4+45050x5+31865x6+15370x7+4840x8+900x9+75x10. In 2013, Guo et al. [9] proved the complete monotonicity on (0,∞) of the function
K(x)=Δ(x)−α(x)900x4(x+1)10. In [10], Zhao et al. proved the complete monotonicity on (0,∞) of the functions
A(x)=−x2+1212x4(x+1)2+Δ(x) and
B(x)=x+1212x4(x+1)−Δ(x). They also presented the double inequality
x2+1212x4(x+1)2<Δ(x)<x+1212x4(x+1),x>0. (2)
The lower bound of inequality (2) and the bound of inequality (1) are not included each other.
In 2015, Qi [11] proved the complete monotonicity on (0,∞) of the function
Δ(x)−x2+βx+1212x4(x+1)2 If and only if β≤0 and so its negative if and only if β≥4. Hence, he deduced the two-sided inequality
x2+δx+1212x4(x+1)2<Δ(x)<x2+λx+1212x4(x+1)2,x>0 (3)
If and only if δ≤0 and λ≥4 and posed a two-sided inequality of the function Δ(x) on x > 0. For δ=0 and λ=4, inequality (3) recovers the lower bound and refines the upper bound of inequality (2).
The survey [12] presented several results about the function Δ(x), its divided difference forms, its variants and its q-analogous including inequalities, positivity, generalizations, (logarithmically) complete monotonicity and applications. Also, the authors pose several open problems. For more results related to this function, we refer to [13–25], and the literature listed therein.
Let the function f (x) be completely monotonic for x∈(0,∞) and denote limx→∞f(x)=f(∞). If the function xβ[f(x)−f(∞)] is completely monotonic on x > 0 if and only if β∈[0,α], then the number α∈R is called the completely monotonic degree [16,26] of f (x) with respect to x > 0 and is denoted by degcmx[f(x)]=α. This concept can help to measure completely monotonic functions more precisely.
Qi [27] proved that
degcmx[Δ(x)]=4 and Guo et al. [26] proved that
degcmx[Δ(x)−C(x)1800x2(x+1)10(x+2)10]=1, where C(x) is a polynomial of positive coefficients of degree twenty one. For more results related to properties of completely monotonic degree, we refer to [12,16,28–32] and the references therein.
Our first aim will be to establish the double inequality
x2+3x+33x4(2x+1)2<Δ(x)<625x2+2275x+50433x4(50x+41)2,x>0 which improves the upper bound of inequality (3) for x > 0 and the lower bound of inequality (3) for x>132(9+849)≈1.1918. Also, we proved the complete monotonicity of the following two functions on (0,∞):
Δ(x)−x2+3x+33x4(2x+1)2 and
625x2+2275x+50433x4(50x+41)2−Δ(x). The second aim of this paper is to compute the completely monotonic degree of a function involving ψ′(x).
2 Lemmas
For proving our main results, we need the following lemmas.
Lemma 2.1 The function
L(x)=ψ′(x)−12x2−1x−x22[x2+3x+33x4(2x+1)2−(x+1)2+3(x+1)+33(x+1)4(2x+3)2] (4)
is completely monotonic on (0,∞) and
ψ′(x)>k(x)6x2(x+1)4(2x+1)2(2x+3)2,x>0, (5)
where
k(x)=96x9+816x8+3040x7+6536x6+8968x5+8157x4+4962x3+1977x2+477x+54. Proof. Using the integral representations formulas [33]
1xs=1Γ(s)∫0∞vs−1e−xvdv,s,x>0 (6)
and
(−1)mψ(r)(x)=∫0∞vme−xve−v−1dv,m∈N,x>0, (7)
we have
L(x)=∫0∞e−3v/224(ev−1)ℓ(v)e−xvdv,x>0, (8)
where
ℓ(x)=−2ev/2(v3−15v2+92v−246)+2e3v/2(v3−15v2+104v−252)−63v+12e5v/2−e2v(7v+36)+ev(70v+528)−492=∑n=5∞anvn2nn! with
an=1627(3n−27)n3−89(17×3n−189)n2+(35×2n+1−7×22n−1+4136×3n−3−520)n+12(11×2n+2−14×3n+1−3×4n+5n+41). Using the inequalities 3n > n2 for n≥1, we get
an[1]=1627(3n−27)n3−89(17×3n−189)n2=n2[3n(16n27−1369)−16n+168]>n2[n2(16n27−1369)−16n]>8n327(2n2−51n−54)>0,n≥27. Also, by using the inequalities 12(54)n>7n2+708 for n≥19 and 2n≥n for n≥1, we have
an[2]=(35×2n+1−7×22n−1+4136×2n−3−520)n+12(11×2n+2−14×4n+1−3×4n+5n+41)>[12×5n−22n(7n2+708)]+[2n(587n+528)−520n]>0,n≥19 and hence an > 0 for n≥27. Furthermore, Mathematica software computation shows that ∑n=526anvn2nn! is a polynomial in v with all positive coefficients. Then an > 0 for n≥5 and hence ℓ(v)>0 for v > 0 and consequently the function L(x) is completely monotonic in (0,∞). Now the function L(x) is decreasing, and using the asymptotic expansion [33]
ψ′(x)∼1x+12x2+∑m=2∞Bmxm+1,x→∞, (9)
where the mth Bernoulli number Bm is defined by [34]
∑m=0∞Bmm!tm=tet−1,|t|<2π, We obtain limx→∞L(x)=0 and hence L(x) > 0 for x > 0.
Lemma 2.2 The function
U(x)=x22[625x2+2275x+50433x4(50x+41)2−625(x+1)2+2275(x+1)+50433(x+1)4(50x+91)2]+1x+12x2−ψ′(x) (10)
is completely monotonic on (0,∞) and
ψ′(x)<x22[625x2+2275x+50433x4(50x+41)2−625(x+1)2+2275(x+1)+50433(x+1)4(50x+91)2]+1x+12x2,x>0. (11)
Proof. Using the formulas (6) and (7), we obtain
U(x)=∫0∞e−91v/5067818264(ev−1)u(v)e−xvdv,x>0 (12)
where
u(v)=e91v/50(5651522v3−67613182v2+354726096v−491362936)+e41v/50(−5651522v3+67613182v2−286907832v+491773100)−1681e2v(14391v+40100)−410164e141v/50−1183(100737v+415700)+2ev(71681571v+279590600)=−∑n=5∞bn2−n−125−nvn753571n! with
bn=(54017153140041×2n+225n−709777372049200×41n+327338438907600×91n−18229840278741×100n)n+10000(19789528031×41n−3500263931×91n)n2−6028568(−1397953×2n+225n+1+674081×4n25n+1−122943275×41n+122840734×91n+102541×141n)−20500000(753571×41n−68921×91n)n3. Now
bn[1]=(−18229840278741×100n+54017153140041×2n+225n)n=52n2nn(−18229840278741×2n+216068612560164)<0,n≥5. By using that (9141)n>19789528031n−70977737204.923500263931n−32733843890.76 for n≥10 we deduce that
bn[2]=(−709777372049200×41n+327338438907600×91n)n+10000(19789528031×41n−3500263931×91n)n2<0,n≥10. For n≥24, n3<(32)n and hence
bn[3]=−6028568(−1397953×2n+225n+1+674081×4n25n+1−122943275×41n+122840734×91n+102541×141n)−20500000(753571×41n−68921×91n)n3<23−n[18839275 41n(4917731 2n−102500 3n)−507967893251 8n25n+1+753571×2n+1(1397953×50n+1−61420367×91n)+1681×3n(105062500×91n−45967831×94n)]<0forn≥26. Hence bn < 0 for n≥26. Using Mathematica, we can see that all coefficients of the polynomial ∑n=525(−bn2−n−125−nvn753571n!) are positive. Then bn < 0 for n≥5, hence u(v) > 0 for v > 0, and consequently the function U(x) is completely monotonic in (0,∞). Now the function U(x) is decreasing, and using the asymptotic expansion (9), we obtain limx→∞U(x)=0 and hence U(x) > 0 for x > 0.
3 Main Results
Now we begin to prove our main results.
Theorem 3.1 The functions
fl(x)=Δ(x)−x2+3x+33x4(2x+1)2 (13)
and
fu(x)=625x2+2275x+50433x4(50x+41)2−Δ(x) (14)
are completely monotonic on (0,∞) and
x2+3x+33x4(2x+1)2<Δ(x)<625x2+2275x+50433x4(50x+41)2,x>0. (15)
Proof. Using recursion formula [33]
ψ(x+1)=ψ(x)+1x,x>0 (16)
we get
fl(x)−fl(x+1)=ψ″(x)−ψ″(x+1)+(x+1)2+3(x+1)+33(x+1)4(2(x+1)+1)2+{ψ′(x)+ψ′(x+1)}{ψ′(x)−ψ′(x+1)}−x2+3x+33x4(2x+1)2=2x2L(x). The two functions 2x2 and L(x) are completely monotonic on (0;∞) and the product of two completely monotonic functions is also completely monotonic, then the difference fl(x) − fl(x + 1) is completely monotonic on (0;∞), and hence the function fl(x) is also completely monotonic on (0;∞), see [11]. Now using the formula (9) and its derivative formula
ψ″(x)∼−1x2−1x3−∑m=2∞(m+1)Bmxm+2,x→∞ (17)
We get limx→∞fl(x)=0 and hence fl(x) > 0 for x > 0, that is
Δ(x)>x2+3x+33x4(2x+1)2,x>0. Now, using the recursion formula (16), we get
fu(x)−fu(x+1)=ψ″(x+1)−ψ″(x)+625x2+2275x+50433x4(50x+41)2+{ψ′(x+1)+ψ′(x)}{ψ′(x+1)−ψ′(x)}−625(x+1)2+2275(x+1)+50433(x+1)4(50(x+1)+41)2=2x2U(x). The two functions 2x2 and U(x) are completely monotonic on (0;∞), and hence the functions fu(x) − fu(x + 1), and fu(x) are completely monotonic on (0;∞). Now using the formulas (9) and (17), we have limx→∞fu(x)=0, and then fu(x) > 0 for x > 0, that is
Δ(x)<625x2+2275x+50433x4(50x+41)2,x>0. The proof of Theorem 1 is complete.
Remark 1. The upper bound of inequality (15) is better than the upper bound of inequality (3) for x > 0. Also, the lower bound of inequality (15) is better than the lower bound of inequality (3) for x>132(9+849)≈1.1918.
Theorem 3.2. The completely monotonic degree of L(x) on (0,∞) is 1.
Proof. Using the integral representation (8), we get
L(x)=1x∫0∞e−3v/248(ev−1)2m(v)e−xvdv,x>0, where
m(v)=8e3v/2(v3−18v2+122v−344)−4ev/2(v3−18v2+122v−338)+ev(385v+2722)−4e5v/2(v3−18v2+134v−350)+e3v(7v+22)−27(7v+50)−e2v(203v+1394)=∑n=5∞kn2−n−1vn3375n! with
kn=22n(82944n2+9450000)[(54)n−68.5125n+940.958.2944n2+945]+6n(1728n3+1528416n)[157.5n+148517.28n3+15284.16n−(56)n]+2n(2598750n+18373500)+103(16n3−480n2+4856n−18576)×[3n−216n3−2592n2+8964n−912616n3−480n2+4856n−18576]. Now using the inequalities
68.5125n+940.958.2944n2+945<(54)nn≥1,157.5n+148517.28n3+15284.16n>(56)nn≥27,216n3−2592n2+8964n−912616n3−480n2+4856n−18576<3nn≥15 we obtain that kn > 0 for n≥27. Furthermore, Mathematica computation shows that ∑n=526kn2−n−1vn3375n! is a polynomial in v with all positive coefficients. Then kn > 0 for n≥5, and hence m(v) > 0 for v > 0 and consequently the function x L(x) is completely monotonic in (0,∞). Hence we have
degcmx[L(x)]≥1. (18)
If we suppose that xμL(x) is completely monotonic on (0,∞), then the function xμL(x) is decreasing, that is
μ≤−xL′(x)L(x)=−x[ψ″(x)−ϕ′(x)]ψ′(x)−ϕ(x), where
ϕ(x)=k(x)6x2(x+1)4(2x+1)2(2x+3)2 with
k(x)=96x9+816x8+3040x7+6536x6+8968x5+8157x4+4962x3+1977x2+477x+54. Using the relation [35]
ψ(r)(x)=(−1)r+1r!∑i=0∞1(i+x)r+1,x>0;r=1,2,3,... We get
limx→0[xψ′(x)−1x]=0andlimx→0[x2ψ″(x)+2x]=0. Also,
limx→0[xϕ(x)−1x]=−12andlimx→0[x2ϕ′(x)+2x]=12. Then
μ≤−[x2ψ″(x)+2x]−[x2ϕ′(x)+2x][xψ′(x)−1x]−[xϕ(x)−1x]→1asx→0 and hence we get
degcmx[L(x)]≤1. (19)
Combining (18) and (19) completes the proof.
4 Conclusions
The main conclusions of this paper are stated in Theorems 3.1 and 3.2. Concretely speaking, the authors proved the completely monotonic property of two functions involving the sum of the Trigamma square and Tetragamma functions, derived a new double inequality for this sum and deduced the completely monotonic degree of a function involving the Trigamma function.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
References
- Andrews, G. E., Askey, R. A., Roy, R. (1999). Special functions, encyclopedia of mathematics and its applications, vol. 71. Cambridge: Cambridge University Press.
- Coffey, M. W. (2006). One integral in three ways: Moments of a quantum distribution. Journal of Physics A: Mathematical and General, 39, 1425-1431. [Google Scholar] [CrossRef]
- Miller, A. R. (2006). Summations for certain series containing the digamma function. Journal of Physics A: Mathematical and General, 39, 3011-3020. [Google Scholar] [CrossRef]
- Wilkins, B. D., & Hromadka, T. V. (2021). Using the digamma function for basis functions in mesh-free computational methods. Engineering Analysis with Boundary Elements, 131, 218-227. [Google Scholar] [CrossRef]
- Qiu, S. L., & Vuorinen, M. (2005). Some properties of the gamma and psi functions, with applications. Mathematics of Computation, 74(250), 723-742. [Google Scholar] [CrossRef]
- Mitrinović, D. S., Pečarić, J. E., Fink, A. M. (1993). Classical and new Inequalities in Analysis, Dordrecht-Boston-London: Kluwer Academic Publishers.
- Widder, D. V. (1946). The laplace transform. Princeton: Princeton University Press.
- Alzer, H. (2004). Sharp inequalities for the digamma and polygamma functions. Forum Mathematicum, 16(2), 181-221. [Google Scholar] [CrossRef]
- Guo, B. N., Zhao, J. L., & Qi, F. (2013). A completely monotonic function involving the tri- and tetra-gamma functions. Mathematica Slovaca, 63(3), 469-478. [Google Scholar] [CrossRef]
- Zhao, J. L., Guo, B. N., & Qi, F. (2012). Complete monotonicity of two functions involving the tri- and tetra-gamma functions. Periodica Mathematica Hungarica, 65(1), 147-155. [Google Scholar] [CrossRef]
- Qi, F. (2015). Complete monotonicity of a function involving the tri- and tetra-gamma functions. Proceedings of the Jangjeon Mathematical Society, 18(2), 253-264. [Google Scholar] [CrossRef]
- Qi, F., & Agarwal, R. P. (2019). On complete monotonicity for several classes of functions related to ratios of gamma functions. Journal of Inequalities and Applications, 2019, 1-42. [Google Scholar] [CrossRef]
- Batir, N. (2007). On some properties of digamma and polygamma functions. Journal of Mathematical Analysis and Applications, 328(1), 452-465. [Google Scholar] [CrossRef]
- Guo, B. N., Qi, F., & Srivastava, H. M. (2010). Some uniqueness results for the non-trivially complete monotonicity of a class of functions involving the polygamma and related functions. Integral Transforms and Special Functions, 21(11), 849-858. [Google Scholar] [CrossRef]
- Guo, B. N., & Qi, F. (2011). A class of completely monotonic functions involving divided differences of the psi and tri-gamma functions and some applications. Journal of the Korean Mathematical Society, 48(3), 655-667. [Google Scholar] [CrossRef]
- Koumandos, S. (2008). Monotonicity of some functions involving the gamma and psi functions. Mathematics of Computation, 77(264), 2261-2275. [Google Scholar] [CrossRef]
- Qi, F. (2014). A completely monotonic function related to the -trigamma function. University Politehnica of Bucharest Scientific Bulletin-Series A–Applied Mathematics and Physics, 76(1), 107-114. [Google Scholar]
- Qi, F. (2015). Complete monotonicity of functions involving the -trigamma and -tetragamma functions. Revista de la Real Academia de Ciencias Exactas, F¡́x0131/¿sicas y Naturales, Serie A Matemáticas, Journal of the Spanish Royal Academy of Sciences, Series A Mathematics,, 109(2), 419-429. [Google Scholar] [CrossRef]
- Qi, F., & Guo, B. N. (2016). Complete monotonicity of divided differences of the di- and tri-gamma functions with applications. Georgian Mathematical Journal, 23(2), 279-291. [Google Scholar] [CrossRef]
- Qi, F., Liu, F. F., & Shi, X. T. (2016). Comments on two completely monotonic functions involving the -trigamma function. Journal of Inequalities and Special Functions, 7(4), 211-217. [Google Scholar]
- Qi, F. (2021). Necessary and sufficient conditions for a difference constituted by four derivatives of a function involving trigamma function to be completely monotonic. Mathematical Inequalities and Applications, 24(3), 845-855. [Google Scholar] [CrossRef]
- Qi, F. (2021). Necessary and sufficient conditions for a ratio involving trigamma and tetragamma functions to be monotonic. Turkish Journal of Inequalities, 5(1), 50-59. [Google Scholar]
- Qi, F. (2021). Necessary and sufficient conditions for complete monotonicity and monotonicity of two functions defined by two derivatives of a function involving trigamma function. Applicable Analysis and Discrete Mathematics, 15(2), 378-392. [Google Scholar] [CrossRef]
- Qi, F. (2021). Necessary and sufficient conditions for two functions defined by two derivatives of a function involving trigamma function to be completely monotonic. TWMS Journal of Pure and Applied Mathematics, 12(2), [Google Scholar]
- Zhao, J. L. (2015). A completely monotonic function relating to the q-trigamma function. Journal of Mathematical Inequalities, 9(1), 53-60. [Google Scholar] [CrossRef]
- Guo, B. N., & Qi, F. (2012). A completely monotonic function involving the tri-gamma function and with degree one. Applied Mathematics and Computation, 218(19), 9890-9897. [Google Scholar] [CrossRef]
- Qi, F. (2020). Completely monotonic degree of a function involving trigamma and tetragamma functions. AIMS Mathematics, 5(4), 3391-3407. [Google Scholar] [CrossRef]
- Guo, B. N., & Qi, F. (2015). On the degree of the weighted geometric mean as a complete Bernstein function. Afrika Matematika, 26(7), 1253-1262. [Google Scholar] [CrossRef]
- Koumandos, S., & Lamprecht, M. (2013). Complete monotonicity and related properties of some special functions. Mathematics of Computation, 82(282), 1097-1120. [Google Scholar] [CrossRef]
- Qi, F., & Wang, S. H. (2014). Complete monotonicity, completely monotonic degree, integral representations, and an inequality related to the exponential, trigamma, and modified Bessel functions. Global Journal of Mathematical Analysis, 2(3), 91-97. [Google Scholar] [CrossRef]
- Qi, F. (2015). Properties of modified Bessel functions and completely monotonic degrees of differences between exponential and trigamma functions. Mathematical Inequalities and Applications, 18(2), 493-518. [Google Scholar] [CrossRef]
- Qi, F., & Liu, A. Q. (2019). Completely monotonic degrees for a difference between the logarithmic and psi functions. Journal of Computational and Applied Mathematics, 361, 366-371. [Google Scholar] [CrossRef]
- Abramowitz, M., Stegun, I. A. (Eds). (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables (9th ed), vol. 55. Washington DC: National Bureau of Standards, Applied Mathematics Series.
- Shuang, Y., Guo, B. N., & Qi, F. (2021). Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios. Revista de la Real Academia de Ciencias Exactas, F¡́x0131/¿sicas y Naturales, Serie A. Matemáticas, 115(3), 1-12. [Google Scholar] [CrossRef]
- Alzer, H. (1997). On some inequalities for the gamma and psi functions. Mathematics of Computation, 66(217), 373-389. [Google Scholar] [CrossRef]