MONOTONICITY AND CONVEXITY PROPERTIES OF THE NIELSEN’S β-FUNCTION Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 6(24), No. 2, 2017, pp. 81-93

DOI: 10.15393/j3.art.2017.3950

81

UDC 517.58

KWARA NANTQMAH

MONOTONICITY AND CONVEXITY PROPERTIES OF THE NIELSEN'S ^-FUNCTION

Abstract. The Nielsen's [-function provides a powerful tool for evaluating and estimating certain integrals, series and mathematical constants. It is related to other special functions such as the digamma function, the Euler's beta function and the Gauss' hypergeometric function. In this work, we prove some mono-tonicity and convexity properties of the function by employing largely the convolution theorem for Laplace transforms.

Key words: Nielsen's [-function, Laplace transform for convolutions, completely monotonic function, convex function, GA-convex function, inequality

2010 Mathematical Subject Classification: 33B99, 26A48

1. Introduction and Preliminaries. Throughout this paper we shall use the following notation: N = {1, 2, 3, 4,... }, No = N U {0} and R = {all real numbers}.

The Nielsen's ^-function ^(x) was introduced in [15] and is defined by any of the following equivalent forms.

1

x > 0,

(1)

0

oo

X > 0,

(2)

0

x > 0,

(3)

©Petrozavodsk State University, 2017

[MglHl

^M'-r) - * (!)} • x> 0- (4)

where *(x) = dx lnr(x) is the digamma or psi function and r(x) is the Euler's Gamma function. It is known that the function P(x) satisfies the following properties [4], [15].

P(x + 1) = ^ - P(x), (5)

x

P(x)+ P(1 - x) = . (6)

sin nx

For additional information on the function refer to [4], [6], [8], [13], and

[14].

Also, the function P(x) is related to the classical Euler's beta function B(x,y) and the Gauss' hypergeometric function 2F1(a, b; c; z) in the following ways.

P (I) =- dx {l°B (f -1

P(x) + P(1 - x) = B(x, 1 - x),

P (x) = i Fi(1,x; x + 1;-1). x2

Repeatedly differentiating (1), (2), (3), (4), and (5) obtain

l

P(m) (x) = J (lnj)mP dj, x > 0 (7)

0

CO

xm „—xt

jm e~

= (-1)^ dj, x> 0 (8)

0

C / -\k

= (-1)mm^7kTxmTI, x> 0 (9)

k=0 (k + x)m+1

1 [,(m)( x + ^ _J,(m)( x

2m+1 r \ 2 V2

(10)

p (m)(x+1) =(xm)+i1 - P (m) (x), (11)

where m e No and ft(0)(x) = ft(x). Note that |ft(m)(x)| = (-1)mft(m)(x) for m e N0 and x > 0. Then, by multiplying the recurrence relation (11) by the factor (— 1)m, we obtain

ft(m)(x +1) = - ft(m) (x) , (12)

xm+1

and as an immediate consequence we obtain the upper bound

ft(m) (x) < ^. (13)

Also, it can easily be shown from (8) that the function |ft(m)(x)| is decreasing in terms of x.

Some special values of the function are given as follows.

ft(1) = ln 2, ft( 2)= 2, ft( 2) =2 - 2, ft (2) = 1 - ln 2,

ft'(D=-2z(2)=-^, ft'<2>=-i+li, ft'(3>=! - ^,

ft' (3) =4(G - * ft' (2) =40 - 4G,

where Z(x) is the Riemann zeta function and G is the Catalan's constant.

As shown in [4] and [8], the Nielsen's ft-function is very useful in evaluating certain integrals. | |

In [1] it was established that the function xc |^(n)(x)|, where n e N, is strictly decreasing (increasing) on (0, to) respectively if c < n (c > n + 1).

The author of [3] established that ) is strictly concave on R, and that ^(xc) is strictly concave (convex) respectively if c > 0 (c e [-1, 0)). The author further established that xc |^(n)(x)|, n e N is strictly convex if and only if c < n, c = n + 1 or c > n + 2.

The authors of [5] showed that ^(k)(ex) is strictly concave (convex) on R if, respectively, k = 2n - 2 (k = 2n - 1), where n e N. They further showed that ^(k)(xc) is convex on (0, to) if either k = 2n - 1 and c e (-to, -2nTr] U (0, to) or k = 2n - 2 and c e [-^nTr, 0).

Also, strict complete monotonicity of x |^(m)(x)|, m e N on (0, to) was established in [9] among other things.

Then the author of [2] proved that An

(x) = xnr |P(n)(x)|, n G N

and An(x) are strictly increasing on (0, to), and that lim An(x) = 1 and lim An (x) =0.

On the account of these results, the natural question is whether similar results can be established for the Nielsen's P-function, since it satisfies some properties identical to those satisfied by the classical digamma function. This is what motivates the present work. We present our findings in the following section.

2. Main Results. We begin by recalling the following well-known definitions which pertain to our results.

Definition 1. A function f : (a, 6) C R ^ R is said to be convex on the interval (a, 6) if

f (Ax + (1 - A)y) < Af (x) + (1 - A)f (y),

holds for all x,y G (a, 6) and A G [0,1]. If f is twice differentiable, then it is said to be convex if and only if f "(x) > 0 for every x G (a, 6). If the inequalities are strict, then f is said to be strictly convex. If the inequalities are reversed, then f is said to be concave.

Definition 2. A function f : (a, 6) C (0, to) ^ R is said to be GA-convex on (a, 6) if

f(xV"A) < Af(x) + (1 - A)f(y), holds for all x, y G (a, 6) and A G [0,1] [16].

Definition 3. A function f : (0, to) ^ R is said to be completely monotonic if f has derivatives of all order and

(-1)kf(k) (x) > 0,

holds for x G (0, to) and k G N0. Theorem 1. Let F be defined as

F(x) = xa P(m) (x) , (14)

where a G R, m G N and x > 0. Then F (x) is decreasing if a < m +1 and increasing if a > m + 1 + e-1.

Proof. By using (8) and the convolution theorem for Laplace transforms, we obtain the following.

= a

where

F'(x) = axa—1 ft(m) (x)

— x

ft (m+1)(x)

=x

F' (x)

xa

a

x

a

x

oo

= aJe

0

sm 1 + e-

—xt

dt

ft (m)(x) ft (m)(x)

tme-xt

1 + e-t

ft(m+1) (x) ft(m+1) (x)

dt -

tm+1e-xt

1 + e

t

dt =

t

ds

xt

dt

£m+1 e—xt

1 + e~ dt = y 0m(t)e—xt dt, 0

0m (t) = a

1 + e-

ds

t

m+1

1 + e

t

Then 0m(O) = lim 0m (t) = 0 and

t—> 0+

0/(t) =

atm

1 + e tm

t

1 + e

t

(m + 1)tm 1 + e—t

a — (m +1) —

tm+1e—t

(1 + e—t)2

te

t

1 + e

t

(15)

If a < m+1, then 0/(t) < 0, which implies that 0m(t) is decreasing. Then for t > 0 we obtain 0m (t) < 0m(O) = 0. Thus, F' (x) < 0 which gives the desired result. Likewise, if a > m + 1 + e—1, then 0/(t) > 0, which implies that 0m (t) is increasing. Then for t > 0, we have 0m (t) > 0m (0) = 0. Hence, F'(x) > 0 and this completes the proof. □

Theorem 2. Let m G N. Then the inequality

ft(m)(xy) < ft(m)(x) + ft(m) (y)

(16)

holds for x > 0 and y > 1.

t

m

s

Proof. Let G(x,y) = |P(m)(xy)| - |P(m)(x)| - |P(m)(y)| for m G N, x > 0 and y > 1. Without loss of generality, let y be fixed. Then

G'(x,y) = -y P(m+1) (xy) + P(m+1)(x) 1

x

x

ß(m+l)(x) - xy ß(m+1)(xy)

Recall from Theorem 1 that the function x |P(m)(x)| is decreasing. Also, since y > 1, then xy > x. Hence, G' (x,y) > 0 and this implies that G(x,y) is increasing. Then for 0 < x < to, we obtain

G(x, y) < lim G(x,y) = - P(m)(y)

which yields the result (16). □

Theorem 3. Let m G N. Then the function

H (x) = x P(m) (x) is strictly completely monotonic on (0, to).

< 0,

Proof. Note that

H' (x) = ß(m) (x) - x ß(m+1) (x)

H''(x) = -2 P(m+1)(x) + x P(m+2)(x)

H''' (x) = 3 P (m+2) (x) - x P (m+3) (x) By continuing the process, we obtain

H(n)(x) = (-1)n-1n p(m+n-1)(x) +(-1)nx P(m+n)(x)

which implies

(—1)nH(n) (x)

n

x

x

ß(m+n-1) (x) + ß(m+n) (x)

(17)

Then by the convolution theorem for Laplace transforms, we obtain

(—1)nH(n) (x)

= — n e

TO CO

— xt

^m+n- le-xt

x

1 + e

t

r tm+n e—xt

dt W ^-— dt

1+e

t

= — n

where

s

m+n—1

1 + e-

ds

e—xt dt +

tm+n e —xt r

1 + e-t dt = 5m(t)e—xt dt,

¿m (t) = n

s

m+n— 1

1 + e-

ds +

tm+n

1 + e—* '

Note that 5m (0) = lim 5m (t) = 0 and also, t^0+

¿m(t) =

tm+n—1 (m + n)tm+n—1 -n—-r + --—2—:-+

1 + e—*

tm+n- 1

1 + e

—t

m +

te

1 + e—* -t

tm+ne—t

(1 + e—* )2

1 + e

—t

> 0.

Hence 5m(t) is increasing. Then for t > 0, we have 5m (t) > 5m(0) = 0. Thus, (— 1)nH(n) (x) > 0 which concludes the proof. □

Theorem 4. Let m G N. Then the function

Q(x) = ft(m) (ax )

is strictly convex on (0, œ). Proof. Direct differentiation yields

Q"(x) = -(ln a)2 |"ax ft(m+1} (ax)

, a > 1,

(18)

a

2x

ft (m+2) (ax )

Let ax = z. Then

Q" (x)

1

(ln a)2z 2 z

ft (m+1)(z) + ft(m+2)(z )

C» C» CO C»

/* /* tm+1e — zt /• tm+2e —zt /•

= - I e—zt dt I \ ^ dt + e I * ^-—zt

1 + e

t

1 + e—T dt = Am(t)e—zt dt,

0

0

0

0

where

Am (t) = -

m+1

1 + e—

m+2

ds +

1+e

t

t

t

t

Then Am (0) = lim Am (t) = 0 and also, t—> 0+

tm+1 (m + 2)tm+1 tm+2e-t

Am(t) = - —-7 + . . , +

1 + e-t

tm+1

1 + e

t

m + 1 +

1 + e-t (1 + e-t)2 te-t

1 + e

t

> 0.

Thus Am(t) is increasing. Then for t > 0, we have Am(t) > Am (0) = 0. Therefore, Q" (x) > 0 and this completes the proof. □

With regard to Theorem 4, there is no a such that Q(x) is concave. Also, Q(x) is increasing if 0 < a < 1 and decreasing if a > 1. Furthermore, the convexity of Q(x) implies that for r, s > 0, u > 1, U + V = 1 we have

P(m) (aU+S"

<

|P (m)(ar )| |p (m)(as )|

+

u

By letting x = ar and y = as we obtain

P(m) fx u y S

< !p(m) (x)1 |p(m)(y)1

u

(19)

which implies that the function |P(m)(x)| is GA-convex. Moreover, since the exponential function is convex, we have

r + s a a a « + v <--1--,

u v

for u > 1, U + 1 = 1. Then by the decreasing property of |P(m) (x)| we obtain

P(mW ^ + O. uv

<

P(m) (a U+S

which upon letting x = ar and y = as, yields

(m) f x + y

uv

<

)

x u yv

Now, combining (19) and (20) gives the double-inequality

P(m) ( X + y ^|<|p (m) ^ i y S )

<

|p(m) (x)| |P (m)(y)|

u

+

v

v

1 1

v

Theorem 5. Let m G N. Then the function

T (x) = ft(m) (xc)

is strictly convex on (0, to) if c < — 1 or c > 0. Proof. Similarly, direct differentiation gives

(21)

T" (x) = —c (c — 1)xc—2 ft(m+1) (xc)

cx

2c—2

ft(m+2) (xc)

= (cxc—1)2 Let xc = z. Then

1 - c 1

1c

e—zt dt

where

Km (t) =

c xc

1 — c 1

c z

tm+1 e -zt

1 + e -t

)=1 - c

ft(m+1)(xc) + ft (m+2)(xc)

ft(m+1) (z) + ft (m+2)(z)

CO

—zt r

— dt = I Km(t)e—zt dt,

r tm+2 e—zt r

dt + 1 I TS zt

1 + e-

0

0

„m+1

y.m+2

c J 1 + e—s

0

ds +

1 + e

t

Then, Km(0) = lim Km(t) = 0 and

t 0+

K/ (t) =

¿m+1

1+e

t

1 , te—t

- + m + 1 + --1

c 1 + e—t

>0

for c < —1 or c > 0. Thus Km (t) is increasing. Then for t > 0 we have Km (t) > K/(0) = 0. Therefore, T" (x) > 0 which yields the desired result. □

Theorem 6. Let a function Dm be defined as

Dm(x) =

x

m+1

m!

ft (m)(x)

(22)

where x > 0 and m G N. Then Dm (x) is strictly decreasing, strictly concave, and the limits

lim Dm (x) = 1 and lim DL (x) = 0 hold.

x 0 x 0 m

c — 1

t

Proof. By direct differentiation we obtain

Dm (x)

(m + 1)xm

m!

P(m)(x)

x

m+1

m!

P(m

x

m+1

m!

m +1

x

P(m)(x) - P(m

(24)

That is,

x

m! , m + 1

m (x)

m+1 '

x

p(m)(x) - p(m+1)(x)

to c»

—xt

= (m + 1) / e dt

^m^—xt

1 + e—t

CO C»

/tm+1g—zt r

1 + e—t dt = Xm(t)e—zt dt,

0

0

where

Xm (t) = (m + 1)

^m tm+1

ds —

1 + e

1 + e

t

Then xm(0) = lim xm(t) = 0 and also, t^0+

tm+1

Xm (t) = - (1 + e-t)2 < 0.

Hence, Xm(t) is decreasing. Then for t > 0 we have xm (t) < Xm (0) = 0. Thus, Dm(x) < 0, which implies that Dm(x) is strictly decreasing. Next, we have Dm (x) =

x

m+1

m!

m(m + 1)

x2

P(m

2(m + 1)

x

p(m

+

P(m

Then,

m! ( ) = m(m + 1)

m+1 Dm (x) = x2

P(m (x

2(m + 1)

x

P(m

+

+

P (m+2)(x)

CO C»

tme—xt

m(m +1W te-xt dt I —-7 dt-

1 + e

t

t

—2(m + 1W e dt

tm+1 e—xt

1+e

t

dt +

tm+2 e—zt

1+e

— dt = ftm(t)e—zt dt,

where

t t /• (t — S)Sm / S fim(t) = m(m + 1W -ds — 2(m + 1)

1 + e—

m+1 tm+2

ds +

1 + e

1 + e

t

Clearly, fim(0) = lim fim(t) = 0. In addition,

t 0+

tfm(t) =

tm+1e—t (1 + e—t)2

tm+1 e—t

(1 + e—t )2

[—m — met + t] =

2m + (m — 1)t + ^

k=2

mtk

"kT

< 0

Hence, Om(t) is decreasing. Then for t > 0 we have (t) < 0m(0) = 0. Thus, Dm (x) < 0, which implies that Dm(x) is strictly concave. Finally, the limits (23) are deduced from (12), (22), and (24). □

3. Concluding Remarks. In this work we have shown that:

(a) F(x) = xa | ft(m) (x) |, m G N is decreasing if a < m + 1 and increasing if a > m + 1 + e—1.

(b) |ft(m)(xy)| < |ft(m)(x)| + |ft(m)(y)|, for x > 0, y > 1 and m G N.

(c) H(x) = x |ft(m)(x)|, m G N is strictly completely monotonic.

(d) Q(x) = |ft(m)(ax)|, m G N, a > 1 is strictly convex.

(e) T(x) = |ft(m)( xc)|, m G N is strictly convex if c < — 1 or c > 0.

<

I^M + i^M, where u > 1,

u 1 v * *

(f) |ft(m) (u + V)| < |ft(m) (xuyv)

1 + 1 =1 and m G N.

U V

(g) Dm(x) = x + |ft(m)(x)|, m G N is strictly decreasing, strictly con-

cave, lim Dm (x) = 1, and lim Dm (x) = 0.

0 x—¥ 0

The function ft(x) studied in this paper satisfies the equality

ft(x) = 1 G(x),

where G(x) is the function referred to as the Bateman's G-function in the works [10], [11], and [12]. In these works their authors made reference to the work of Erdelyi et al. [7] where, probably, the name "Bateman's G-function" originates. However, before the work [7] the function P(x) appeared in Nielsen's work [15]. Also, it is worth noting that page 54 of the work [7] actually captures Nielsen's work in the reference list. As a result of this, we prefer to call either of the functions the Nielsen's P-function.

4. Open Problems.

1. In relation to Theorem 3, find all values of a G R such that the function Ha(x) = xa |p(m) (x)| is completely monotonic.

2. With regard to Theorem 5, is the function T(x) = |P(m) (xc)| concave on (0, to) if c G (-1, 0)?

Acknowledgment. The author is very grateful to the anonymous referees for their useful comments and suggestions, which helped in improving the quality of this paper.

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Received August 27, 2017.

In revised form, November 24, 2017.

Accepted November 28, 2017.

Published online December 27, 2017.

University for Development Studies, Navrongo Campus P. O. Box 24, Navrongo, Upper East Region, Ghana E-mail: knantomah@uds.edu.gh