Some properties of several functions involving polygamma functions and originating from the sectional curvature of the b
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Some properties of several functions involving polygamma functions and originating from the sectional curvature of the beta manifold Feng Qi1,2,3 Accepted: 3 October 2020 © Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract In the paper, the author investigates some properties, including analyticity, limits, monotonicity, complete monotonicity, and inequalities, of several functions involving the tri-, tetra-, and penta-gamma functions and originating from computation of the sectional curvature of the beta manifold. Keywords Property · Polygamma function · Analyticity · Monotonicity · Complete monotonicity · Limit · Inequality · Trigamma function · Tetragamma function · Pentagamma function · Sectional curvature · Beta manifold Mathematics Subject Classification 26A48 · 26A51 · 33B15 · 44A10
1 Motivations and main results Let M = {(x, y) ∶ x, y > 0} denote the first quadrant on ℝ2 . Let
ds2 = 𝜓 � (x)dx2 + 𝜓 � (y)dy2 − 𝜓 � (x + y)(dx + dy)2 be the Fisher metric and M be equipped with ds2 , where
Communicated by Claudio Gorodski. * Feng Qi [email protected]; [email protected]; [email protected] https://qifeng618.wordpress.com 1
College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China
2
Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China
3
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
𝜓(z) = [ln Γ(z)]� =
Γ� (z) Γ(z)
and
Γ(z) =
∫0
∞
tz−1 e−t dt
for ℜ(z) > 0 . In the literature [1, Sect. 6.4], the function Γ(z) is known as the Euler gamma function, the functions 𝜓(z) , 𝜓 � (z) , 𝜓 �� (z) , 𝜓 ��� (z) , and 𝜓 (4) (z) are known as the di-, tri-, tetra-, penta-, and hexa-gamma functions respectively, and, as a whole, all the derivatives 𝜓 (k) (z) for k ≥ 0 are known as the polygamma functions. Proposition 3 in [8, Sect. 2.4] and Proposition 13 in [9] read ( that the ) sectional curvature K(x, y) of the Fisher metric ds2 on the beta manifold M, ds2 is given by [ � ] 𝜓 � (y) 𝜓 � (x+y) 𝜓 �� (x)𝜓 �� (y)𝜓 �� (x + y) 𝜓𝜓��(x) + − �� �� (x) 𝜓 (y) 𝜓 (x+y) (1.1) K(x, y) = . 4[𝜓 � (x)𝜓 � (x + y) + 𝜓 � (y)𝜓 � (x + y) − 𝜓 � (x)𝜓 � (y)]2 Proposition 4 in [8, Sect. 2.4] and Proposition 14 in [9] state that the asymptotic behavior of the sectional curvature K(x, y) is given by ( ) 1 3 𝜓 � (x)𝜓 ��� (x) − lim+ K(x, y) = lim+ K(y, x) = , (1.2) y→0 y→0 2 2 [𝜓 �� (x)]2
lim K(x, y) = lim K(y, x) =
y→∞
y→∞
lim
(x,y)→(0+ ,0+ )
lim
K(x, y) =0,
(x,y)→(0+ ,∞)
K(x, y) =
𝜓 � (x) + x𝜓 �� (x) , 4[x𝜓 � (x) − 1]2
lim
(x,y)→(∞,∞)
lim
(x,y)→(∞,0+ )
1 K(x, y) = − , 2
1 K(x, y) = − . 4
(1.3) (1.4) (1.5)
Recall from [12, Chapter XIII], [24, Chapter 1], and [25, Chapter IV] that, if a function h(t) on an interval I has derivatives of all orders on I and (−1)n h(n) (t) ≥ 0 for t ∈ I and n ∈ {0} ∪ ℕ , then we call h(t) a completely monotonic function on I. Theorem 12b in [25, p. 16
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