In 2006, Koumandos [56] presented a simpler proof of complete monotonicity of the functions (R_{m}(x)).
Since the complete monotonicity of the functions Δ s, t ( x ) and − Δ s, t ( x ) mean the positivity and negativity of the function Δ s, t ( x ), an alternative proof of Theorem 2 was provided once again.
Later, Qi and Guo in [16], Theorem 1, [17], Theorem 1, proved another result concerning the logarithmically complete monotonicity of the functions (xmapsto F_{a,b,c} ( x ) ) and (xmapsto1/F_{a,b,c} ( x ) ) for (x>-min ( a,b,c ) ), where (c=c ( a,b ) ) is a constant depending on a and b.
Qi in [14], Theorem 1, [15], Theorem 1, investigated the logarithmically complete monotonicity of the functions xmapsto F_{a,b,c} ( x ) =left { textstylebegin{array}l@{quad}l} ( frac{Gamma ( x+b ) }{Gamma ( x+a ) } ) ^{1/ ( a-b ) }e^{psi ( x+c ) }, & mbox{if }aneq b, e^{psi ( x+c ) -psi ( x+a ) }, & mbox{if }a=b, end{array}displaystyle right.
for u > 0. This inequality has been generalized in [22] to the complete monotonicity of a function involving divided differences of the digamma and trigamma functions as follows.
Additionally, Qi's another result involving the logarithmically complete monotonicity of the function (m_{s,t}) can be also found in [25, Theorem 1].
Remark 19 It is easy to see that inequality (41) extends and improves inequalities (10), (57), and (58) if s = 1 2. Remark 20 The left-hand side inequality in (42) is better than the corresponding one in (22) but worse than the corresponding one in (15) for n ≥ 2. Remark 21 Formula (37) implies the complete monotonicity of the function θ ( x ) defined by (38) on ( − 1 2, ∞ ).
Ismail et al. [5, 6] further realized that these inequalities are also the consequences of complete monotonicity of such gamma functions' ratios.
Later, Qi and Guo [22, Theorem 1] obtained a generalization of Theorem B by establishing the logarithmically complete monotonicity of the following function: m_{s,t} ( x ) =frac{1}{exp [ psi ( x+theta ( s,t ) ) ] } biggl[ frac{Gamma ( x+t ) }{Gamma ( x+s ) } biggr] ^{1/ ( t-s ) } (1.4) for (x>-min ( s,t,theta ( s,t ) ) ) with (sneq t), and they derived the following.
The complete monotonicity of the q-analogue of the function δ 0, 0 defined by (93) was researched in [73, 74].
