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In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals.

Abstract In this paper, we present four new Windschitl type approximation formulas for the gamma function. By some unique ideas and techniques, we prove that four functions combined with the gamma function and Windschitl type approximation formulas have good properties, such as monotonicity and convexity. These not only yield some new inequalities for the gamma and factorial functions, but also provide a new proof of known inequalities and strengthen known results.

Abstract In this paper, we present a very accurate approximation for the gamma function: Γ(x+1)∼2πx(xe)x(xsinh1x)x/2exp(73241x3(35x2+33))=W2(x) $$ \Gamma( x+1 ) \thicksim\sqrt{2\pi x} \biggl( \frac{x}{e} \biggr) ^{x} \biggl( x\sinh\frac{1}{x} \biggr) ^{x/2}\exp \biggl( \frac{7}{324}\frac{1}{x^{3} ( 35x^{2}+33 ) } \biggr) =W_{2} ( x ) $$ as x→∞ $x\rightarrow\infty$, and we prove that the function x↦lnΓ(x+1)−lnW2(x) $x\mapsto\ln \Gamma ( x+1 ) -\ln W_{2} ( x ) $ is strictly decreasing and convex from (1,∞) $( 1,\infty ) $ onto (0,β) $( 0,\beta ) $, where β=22,02522,032−ln2πsinh1≈0.00002407. $$ \beta=\frac{22{,}025}{22{,}032}-\ln\sqrt{2\pi\sinh1}\approx0.00002407. $$

Abstract Let Wv(x)=xIv(x)/Iv+1(x) $W_{v} ( x ) =xI_{v} ( x ) /I_{v+1} ( x ) $ with Iv $I_{v}$ be the modified Bessel functions of the first kind of order v. In this paper, we prove the monotonicity of the function x↦(Wv(x)−p)2−(2v+2−p)2x2 $$ x\mapsto\frac{ ( W_{v} ( x ) -p ) ^{2}- ( 2v+2-p ) ^{2}}{x^{2}} $$ on (0,∞) $( 0,\infty ) $ for different values of parameter p with v>−2 $v>-2 $. As applications, we deduce some new Simpson–Spector-type inequalities for Wv(x) $W_{v} ( x ) $ and derive a new type of bounds p+rx2+q2 $p+r\sqrt {x^{2}+q^{2}}$ ( r>0 $r>0$) for Wv(x) $W_{v} ( x ) $. In particular, we show that the upper bound Uv−1(2)(x) $U_{v-1}^{ ( 2 ) } ( x ) $ for Wv(x) $W_{v} ( x ) $ is the minimum over all upper bounds {Up(2)(x):p≤v−1,v>−2} $\{ U_{p}^{ ( 2 ) } ( x ) :p\leq v-1,v>-2 \} $, where Up(2)(x)=p+2v+2−pv+2x2+(2v+2−p)2, $$ U_{p}^{ ( 2 ) } ( x ) =p+\sqrt{\frac {2v+2-p}{v+2}x^{2}+ ( 2v+2-p ) ^{2}}, $$ and is not comparable with other sharpest upper bounds. We also find such type of upper bounds for v−1−2 $v>-2$ and for 2v+2

Abstract Among various approximation formulas for the gamma function, Smith showed that Γ(x+12)∼S(x)=2π(xe)x(2xtanh12x)x/2,x→∞, $$ \Gamma \biggl( x+\frac{1}{2} \biggr) \thicksim S ( x ) =\sqrt{2 \pi } \biggl( \frac{x}{e} \biggr) ^{x} \biggl( 2x\tanh \frac{1}{2x} \biggr) ^{x/2}, \quad x\rightarrow \infty , $$ which is a little-known but accurate and simple one. In this note, we prove that the function x↦lnΓ(x+1/2)−lnS(x) $x\mapsto \ln \Gamma ( x+1/2 ) - \ln S ( x ) $ is strictly increasing and concave on (0,∞) $( 0,\infty ) $, which shows that Smith’s approximation is just an upper one.

Abstract In this paper, by using the monotonicity rule for the ratio of two Laplace transforms, we prove that the function x ↦ 1 24 x ( ln Γ ( x + 1 / 2 ) − x ln x + x − ln 2 π ) + 1 − 120 7 x 2 $$ x\mapsto \frac{1}{24x ( \ln \Gamma ( x+1/2 ) -x\ln x+x- \ln \sqrt{2\pi } ) +1}-\frac{120}{7}x^{2} $$ is strictly increasing from ( 0 , ∞ ) $( 0,\infty ) $ onto ( 1 , 1860 / 343 ) $( 1,1860/343 ) $ . This not only yields some known and new inequalities for the gamma function, but also gives some completely monotonic functions related to the gamma function.

Abstract 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.

Abstract In the article, we prove that the double inequality x 2 + p 0 x + p 0 < Γ ( x + 1 ) < x 2 + 9 / 5 x + 9 / 5 $$ \frac{x^{2}+p_{0}}{x+p_{0}}< \Gamma(x+1)< \frac{x^{2}+9/5}{x+9/5} $$ holds for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , we present the best possible constants λ and μ such that λ ( x 2 + 9 / 5 ) x + 9 / 5 ≤ Γ ( x + 1 ) ≤ μ ( x 2 + p 0 ) x + p 0 $$ \frac{\lambda(x^{2}+9/5)}{x+9/5}\leq\Gamma(x+1)\leq\frac{\mu (x^{2}+p_{0})}{x+p_{0}} $$ for all x ∈ ( 0 , 1 ) $x\in(0, 1)$ , and we find the value of x ∗ $x^{\ast}$ in the interval ( 0 , 1 ) $(0, 1)$ such that Γ ( x + 1 ) > ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)>(x^{2}+1/\gamma)/(x+1/\gamma)$ for x ∈ ( 0 , x ∗ ) $x\in(0, x^{\ast})$ and Γ ( x + 1 ) < ( x 2 + 1 / γ ) / ( x + 1 / γ ) $\Gamma(x+1)

Abstract In the article, we provide a monotonicity rule for the function [ P ( x ) + A ( x ) ] / [ P ( x ) + B ( x ) ] $[P(x)+A(x)]/[P(x)+B(x)]$ , where P ( x ) $P(x)$ is a positive differentiable and decreasing function defined on ( − R , R ) $(-R, R)$ ( R > 0 $R>0$ ), and A ( x ) = ∑ n = n 0 ∞ a n x n $A(x)=\sum ^{\infty}_{n=n_{0}}a_{n}x^{n}$ and B ( x ) = ∑ n = n 0 ∞ b n x n $B(x)=\sum^{\infty }_{n=n_{0}}b_{n}x^{n}$ are two real power series converging on ( − R , R ) $(-R, R)$ such that the sequence { a n / b n } n = n 0 ∞ $\{a_{n}/b_{n}\}_{n=n_{0}}^{\infty}$ is increasing (decreasing) with a n 0 / b n 0 ≥ ( ≤ ) 1 $a_{n_{0}}/b_{n_{0}}\geq(\leq)\ 1$ and b n > 0 $b_{n}>0$ for all n ≥ n 0 $n\geq n_{0}$ . As applications, we present new bounds for the complete elliptic integral E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 t d t $\mathcal{E}(r)=\int_{0}^{\pi /2}\sqrt{1-r^{2}\sin^{2}t}\,dt$ ( 0 < r < 1 $0< r

Abstract In the article, we present new bounds for the function e t cot ( t ) − 1 $e^{t\cot(t)-1}$ on the interval ( 0 , π / 2 ) $(0, \pi/2)$ and find sharp estimations for the Sine integral and the Catalan constant based on a new monotonicity criterion for the quotient of power series, which refine the Redheffer and Becker-Stark type inequalities for tangent function.

Abstract In the article, we prove that the double inequalities π e − x 2 ( x + a ) < K 0 ( x ) < π e − x 2 ( x + b ) , 1 + 1 2 ( x + a ) < K 1 ( x ) K 0 ( x ) < 1 + 1 2 ( x + b ) $$ \frac{\sqrt{\pi}e^{-x}}{\sqrt{2(x+a)}}< K_{0}(x)< \frac{\sqrt{\pi }e^{-x}}{\sqrt{2(x+b)}},\qquad 1+ \frac{1}{2(x+a)}< \frac {K_{1}(x)}{K_{0}(x)}< 1+\frac{1}{2(x+b)} $$ hold for all x > 0 $x>0$ if and only if a ≥ 1 / 4 $a\geq1/4$ and b = 0 $b=0$ if a , b ∈ [ 0 , ∞ ) $a, b\in[0, \infty)$ , where K ν ( x ) $K_{\nu}(x)$ is the modified Bessel function of the second kind. As applications, we provide bounds for K n + 1 ( x ) / K n ( x ) $K_{n+1}(x)/K_{n}(x)$ with n ∈ N $n\in\mathbb{N}$ and present the necessary and sufficient condition such that the function x ↦ x + p e x K 0 ( x ) $x\mapsto\sqrt {x+p}e^{x}K_{0}(x)$ is strictly increasing (decreasing) on ( 0 , ∞ ) $(0, \infty)$ .

Abstract In the article, we provide the necessary and sufficient conditions for the parameters α and β such that the generalized Wilker-type inequality 2 β α + 2 β ( sin x x ) α + α α + 2 β ( tan x x ) β − 1 > ( < ) 0 $$ \frac{2\beta}{\alpha+2\beta} \biggl(\frac{\sin x}{x} \biggr)^{\alpha}+ \frac{\alpha}{\alpha+2\beta} \biggl(\frac{\tan x}{x} \biggr)^{\beta}-1>(< )0 $$ holds for all x ∈ ( 0 , π / 2 ) $x\in(0, \pi/2)$ .

Abstract In the article, we present the necessary and sufficient condition for the parameter p on the interval ( 7 / 5 , ∞ ) $(7/5, \infty)$ such that the function x → erf ( x ) / B p ( x ) $x\rightarrow\operatorname{erf}(x)/B_{p}(x)$ is strictly increasing (decreasing) on ( 0 , ∞ ) $(0, \infty)$ , and find the best possible parameters p, q on the interval ( 7 / 5 , ∞ ) $(7/5, \infty)$ such that the double inequality B p ( x ) < erf ( x ) < B q ( x ) $B_{p}(x) 0 $x>0$ , where erf ( x ) = 2 ∫ 0 x e − t 2 d t / π $\operatorname{erf}(x)=2\int_{0}^{x}e^{-t^{2}}\,dt/\sqrt{\pi}$ is the error function, B p ( x ) = 1 − λ ( p ) e − p x 2 − [ 1 − λ ( p ) ] e − μ ( p ) x 2 $B_{p}(x)=\sqrt{1-\lambda(p)e^{-px^{2}}-[1-\lambda(p)]e^{-\mu(p)x^{2}}}$ , λ ( p ) = 16 ( 5 p − 7 ) / [ ( 15 p 2 − 40 p + 28 ) ( 45 p 2 − 60 p − 4 ) ] $\lambda(p)=16(5p-7)/[(15p^{2}-40p+28)(45p^{2}-60p-4)]$ and μ ( p ) = 4 ( 5 p − 7 ) / [ 5 ( 3 p − 4 ) ] $\mu(p)=4(5p-7)/[5(3p-4)]$ .

Abstract In the article, we discuss the monotonicity properties of the function x → ( 1 − e − a x p ) 1 / p / ∫ 0 x e − t p d t $x\rightarrow (1-e^{-ax^{p}} )^{1/p}/\int_{0}^{x}e^{-t^{p}}\,dt$ for a , p > 0 $a, p>0$ with p ≠ 1 $p\neq1$ on ( 0 , ∞ ) $(0, \infty)$ and prove that the double inequality Γ ( 1 + 1 / p ) ( 1 − e − a x p ) 1 / p < ∫ 0 x e − t p d t < Γ ( 1 + 1 / p ) ( 1 − e − b x p ) 1 / p $\Gamma(1+1/p) (1-e^{-a x^{p}} )^{1/p} 0 $x>0$ if and only if a ≤ min { 1 , Γ − p ( 1 + 1 / p ) } $a\leq\min\{1, \Gamma^{-p}(1+1/p)\}$ and b ≥ max { 1 , Γ − p ( 1 + 1 / p ) } $b\geq\max\{1, \Gamma^{-p}(1+1/p)\}$ .

Abstract Let ψ n = ( − 1 ) n − 1 $\psi_{n}= ( -1 ) ^{n-1}$ ψ ( n ) $\psi^{ ( n ) }$ ( n = 0 , 1 , 2 , … $n=0,1,2,\ldots $ ), where ψ ( n ) $\psi^{ ( n ) }$ denotes the psi and polygamma functions. We prove that for n ≥ 0 $n\geq0$ and two different real numbers a and b, the function x ↦ ψ n − 1 ( ∫ a b ψ n ( x + t ) d t b − a ) − x $$ x\mapsto\psi_{n}^{-1} \biggl( \frac{\int_{a}^{b}\psi_{n}(x+t)\,dt}{b-a} \biggr) -x $$ is strictly increasing from ( − min ( a , b ) , ∞ ) $( -\min ( a,b ) ,\infty ) $ onto ( min ( a , b ) , ( a + b ) / 2 ) $( \min ( a,b ) , ( a+b ) /2 ) $ , which generalizes a well-known result. As an application, the complete monotonicity for a ratio of gamma functions is improved.

Abstract In the article, we deal with the monotonicity of the function x → [ ( x p + a ) 1 / p − x ] / I p ( x ) $x\rightarrow[ (x^{p}+a )^{1/p}-x]/I_{p}(x)$ on the interval ( 0 , ∞ ) $(0, \infty)$ for p > 1 $p>1$ and a > 0 $a>0$ , and present the necessary and sufficient condition such that the double inequality [ ( x p + a ) 1 / p − x ] / a < I p ( x ) < [ ( x p + b ) 1 / p − x ] / b $[ (x^{p}+a )^{1/p}-x]/a< I_{p}(x) 0 $x>0$ and p > 1 $p>1$ , where I p ( x ) = e x p ∫ x ∞ e − t p d t $I_{p}(x)=e^{x^{p}}\int_{x}^{\infty}e^{-t^{p}}\,dt$ is the incomplete gamma function.

Abstract In the article, we establish several inequalities for the Ramanujan constant function R ( x ) = − 2 γ − ψ ( x ) − ψ ( 1 − x ) $R(x)=-2\gamma-\psi(x)-\psi(1-x)$ on the interval ( 0 , 1 / 2 ] $(0, 1/2]$ , where ψ ( x ) $\psi(x)$ is the classical psi function and γ = 0.577215 ⋯ $\gamma=0.577215\cdots$ is the Euler-Mascheroni constant.

Abstract In this paper, we present the monotonicity and absolute monotonicity properties for the two-parameter hyperbolic and trigonometric functions. As applications, we find several complete monotonicity properties for the functions involving the gamma function and provide the bounds for the error function.

Abstract In the article, we present certain p , q ∈ R $p, q\in\mathbb{R}$ such that the Wilker-type inequalities 2 q p + 2 q ( sin x x ) p + p p + 2 q ( tan x x ) q > ( < ) 1 and ( π 2 ) p ( sin x x ) p + [ 1 − ( π 2 ) p ] ( tan x x ) q > ( < ) 1 $$\begin{aligned}& \frac{2q}{p+2q} \biggl(\frac{\sin x}{x} \biggr)^{p}+ \frac{p}{p+2q} \biggl(\frac{\tan x}{x} \biggr)^{q}>(< )1\quad \mbox{and}\\& \biggl(\frac{\pi}{2} \biggr)^{p} \biggl(\frac{\sin x}{x} \biggr)^{p}+ \biggl[1- \biggl(\frac{\pi}{2} \biggr)^{p} \biggr] \biggl(\frac{\tan x}{x} \biggr)^{q}>(< )1 \end{aligned}$$ hold for all x ∈ ( 0 , π / 2 ) $x\in(0, \pi/2)$ .

Abstract In the article, we prove that the function r ↦ E ( r ) / S 9 / 2 − p , p ( 1 , r ′ ) $r\mapsto \mathcal{E}(r)/S_{9/2-p, p}(1, r')$ is strictly increasing on ( 0 , 1 ) $(0, 1)$ for p ≤ 7 / 4 $p\leq7/4$ and strictly decreasing on ( 0 , 1 ) $(0, 1)$ for p ∈ [ 2 , 9 / 4 ] $p\in [2, 9/4]$ , where r ′ = 1 − r 2 $r'=\sqrt{1-r^{2}}$ , E ( r ) = ∫ 0 π / 2 1 − r 2 sin 2 ( t ) d t $\mathcal{E}(r)=\int_{0}^{\pi/2}\sqrt{1-r^{2}\sin^{2}(t)}\,dt$ is the complete elliptic integral of the second kind, and S p , q ( a , b ) = [ q ( a p − b p ) / ( p ( a q − b q ) ) ] 1 / ( p − q ) $S_{p, q}(a, b)=[q(a^{p}-b^{p})/(p(a^{q}-b^{q}))]^{1/(p-q)}$ is the Stolarsky mean of a and b. As applications, we present several new bounds for E ( r ) $\mathcal{E}(r)$ , the Toader mean T ( a , b ) = ( 2 / π ) ∫ 0 π / 2 a 2 cos 2 t + b 2 sin 2 t d t ${T}(a,b)=(2/\pi)\int_{0}^{\pi/2}\sqrt{a^{2}\cos^{2}t+b^{2}\sin^{2}t}\,dt$ , and the Toader-Qi mean TQ ( a , b ) = ( 2 / π ) ∫ 0 π / 2 a cos 2 θ b sin 2 θ d θ $\operatorname{TQ}(a,b)=(2/\pi)\int_{0}^{\pi/2}a^{\cos^{2}\theta}b^{\sin^{2}\theta }d\theta$ .

Abstract In the article, we present several sharp bounds for the modified Bessel function of the first kind I 0 ( t ) = ∑ n = 0 ∞ t 2 n 2 2 n ( n ! ) 2 $I_{0}(t)=\sum_{n=0}^{\infty}\frac{t^{2n}}{2^{2n}(n!)^{2}}$ and the Toader-Qi mean T Q ( a , b ) = 2 π ∫ 0 π / 2 a cos 2 θ b sin 2 θ d θ $TQ(a,b)=\frac{2}{\pi}\int_{0}^{\pi/2}a^{\cos^{2}\theta }b^{\sin^{2}\theta}\,d\theta$ for all t > 0 $t>0$ and a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ .

Abstract In this paper, we present the best possible parameters p and q such that the double inequality M p ( a , b ) < V ( a , b ) < M q ( a , b ) $$ M_{p}(a,b)< V(a,b)< M_{q}(a,b) $$ holds for all a , b > 0 $a, b>0$ with a ≠ b $a\neq b$ , where M r ( a , b ) = [ ( a r + b r ) / 2 ] 1 / r $M_{r}(a,b)=[(a^{r}+b^{r})/2]^{1/r}$ ( r ≠ 0 $r\neq0$ ) and M 0 ( a , b ) = a b $M_{0}(a,b)= \sqrt {ab}$ is the rth power mean and V ( a , b ) = ( a − b ) / [ 2 sinh − 1 ( ( a − b ) / 2 a b ) ] $V(a,b)=(a-b)/[\sqrt{2}\sinh^{-1}((a-b)/\sqrt{2ab})]$ is the second Yang mean.

Let I v x be he modified Bessel function of the first kind of order v. We prove the double inequality sinh t t cosh 1 / q q t < I 0 t < sinh t t cosh 1 / p p t holds for t > 0 if and only if p ≥ 2 / 3 and q ≤ ln 2 / ln π . The corresponding inequalities for means improve already known results.

Let K ( r ) be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for K ( r ) . More precisely, we establish the inequality 2 π K ( r ) > 5 ( r ′ ) 2 + 126 r ′ + 61 61 ( r ′ ) 2 + 110 r ′ + 21 for r ∈ ( 0 , 1 ) , where r ′ = 1 − r 2 . The lower bound is sharp.

We prove that the inequality TQ(a,b)>Lp(a,b) holds for all a,b>0 with a≠b if and only if p≤3/2, where TQ(a,b)=2/π∫0π/2acos2θbsin2θdθ, Lp(a,b)=[(bp-ap)/(p(b-a))]1/p (p≠0), and L0(a,b)=ab are, respectively, the Toader-Qi and p-order logarithmic means of a and b. As applications, we find two fine inequalities chains for certain bivariate means.

We prove that the double inequality Mp(a,b)0 with a≠b if and only if p≤1/3 and q≥log 2/(1+log 2)=0.4093…, where X(a,b) and Mr(a,b) are the Sándor and rth power means of a and b, respectively.

We present the best possible parameters p, q∈0,∞ such that the double inequality 1/3p2cosh(px)+1-1/3p2

We present the necessary and sufficient condition for the monotonicity of the ratio of the power and second Seiffert means. As applications, we get the sharp upper and lower bounds for the second Seiffert mean in terms of the power mean.

We establish several sharp inequalities for trigonometric functions and present their corresponding inequalities for bivariate means.

We present the best possible parameters p and q such that the double inequality (2/3)cos2p(t/2)+1/31/p

We improve the Jordan, Adamović-Mitrinović, and Cusa inequalities. As applications, several new Shafer-Fink type inequalities for inverse sine function and bivariate means inequalities are established, and a new estimate for sine integral is given.

We prove that p=1 and q=2 are the best possible parameters in the interval (0,∞) such that the double inequality ep/x+1-e-p/x/2p0. As applications, some new approximation algorithms for the circumference ratio π and Catalan constant G=∑n=0∞-1n/(2n+1)2 are given. Here, ψ′ is the trigamma function.

We continue to adopt notations and methods used in the papers illustrated by Yang (2009, 2010) to investigate the monotonicity properties of the ratio of mixed two-parameter homogeneous means. As consequences of our results, the monotonicity properties of four ratios of mixed Stolarsky means are presented, which generalize certain known results, and some known and new inequalities of ratios of means are established.

The necessary and sufficient conditions for Schur geometrical convexity of the four-parameter means are given. This gives a unified treatment for Schur geometrical convexity of Stolarsky and Gini means.

Applying well properties of homogeneous functions, some monotonicity results for the ratio of two-parameter symmetric homogeneous functions are presented, which give an easier access to find two-parameter symmetric homogeneous means having ratio simple monotonicity properties proposed by L. Losonczi. As an application, a chain of inequalities of ratio of bivariate means is established.

A four-parameter homogeneous mean ðÂÂÂ…(p,q;r,s;a,b) is defined by another approach. The criterion of its monotonicity and logarithmically convexity is presented, and three refined chains of inequalities for two-parameter mean values are deduced which contain many new and classical inequalities for means.

Let a,b,c,d ∈ R with a+b = c+d = 2r +1. Then In Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) ~ ∞/Σ/k=1 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 as x→∞, where (δ1,δ2) = (|a−b|,|c−d|) = (1−2θ1,1−2θ2). When 0 ≤ δ2 < δ1 ≤ 1, the function x→ (-1)m [1n Γ(x+a) Γ (x+b)/Γ(x+c) Γ (x+d) - m/Σ/k=1 B2k (θ1) - B2k (θ2)/k (2k-1) (x+r)2k-1 ] for m ∈ N is completely monotonic on (−r,∞). This yields some known and new results. [ABSTRACT FROM AUTHOR]

For r, s ∈ R and ρ = min {r, s}, let D [x + r, x + s; ψn-1] ≡ -φn (x) be the divided difference of the functions ψn-1 = (-1)n ψ(n-1) (n ∈ N) on (-ρ, ∞), where ψ(n) stands for the polygamma functions. In this paper, we present the necessary and sufficient conditions for the functions... to be completely monotonic on (-ρ, ∞), where mi, ni ∈ N for i = 1, ..., k with k ≥ 2 and .... These generalize known results and gives an answer to a problem. [ABSTRACT FROM AUTHOR]