MONOTONICITY AND LOGARITHMIC CONVEXITY

MONOTONICITY AND LOGARITHMIC CONVEXITY RELATING TO THE VOLUME OF THE UNIT BALL BAI-NI GUO AND FENG QI Abstract. Let Ωn denote the volume of the unit ball in Rn for n ∈ N. In the 1/(n ln n) is logarithmically convex present paper, we prove that the sequence Ωn and that the sequence

1/(n ln n) Ωn 1/[(n+1) ln(n+1)]

Ωn+1

is strictly decreasing for n ≥ 2. In

addition, some monotonic and concave properties of several functions relating to Ωn are extended and generalized.

1. Introduction In [5, Lemma 2.39], the following result was obtained: The function 1 x ln Γ 1 + x 2 is strictly increasing from [2, ∞) onto [0, ∞) and x 1 1 ln Γ 1 + lim = , x→∞ x ln x 2 2 where

Z

(1)

(2)

∞

tx−1 e−t d t

Γ(x) =

(3)

0

for x > 0 denotes the classical Euler gamma function Γ(x). From this, the following 1/n conclusions were deduced in [5, Lemma 2.40]: The sequence Ωn decreases strictly P∞ 1/ ln n to 0 as n → ∞, the series n=2 Ωn is convergent, and limn→∞ an = e−1/2 , where ln n) an = Ω1/(n (4) n and π n/2 Ωn = (5) Γ(1 + n/2) stands for the n-dimensional volume of the unit ball Bn in Rn . Further, it was conjectured in [5, Remark 2.41] that the function in (2) is strictly increasing from [2, ∞) onto 0, 21 and this would imply that the sequence an is strictly decreasing for n ≥ 2. 2010 Mathematics Subject Classification. Primary 26A48, 33B15; Secondary 26A51, 26D07. Key words and phrases. Monotonicity; logarithmical convexity; volume; unit ball; inequality; gamma function. The second author was partially supported by the China Scholarship Council and the Science Foundation of Tianjin Polytechnic University. Please cite this article as “Bai-Ni Guo and Feng Qi, Monotonicity and logarithmic convexity relating to the volume of the unit ball, Optimization Letters 7 (2013), no. 6, 1139–1153; Available online at https://doi.org/10.1007/s11590-012-0488-2.” 1

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B.-N. GUO AND F. QI

In [3, Theorem 1.5], it was proved that the function ln Γ(x + 1) (6) x ln x is strictly increasing from (1, ∞) onto (1 − γ, 1), where γ is the Euler-Mascheroni constant. From this, the above-mentioned conjecture in [5, Remark 2.41] were resolved in [3, Corollary 3.1]. In addition, it was conjectured in [3, Conjecture 3.3] that the function (6) is concave on (1, ∞). In [12] and [17, Theorem 4], the function (6) was proved to be strictly increasing on (0, ∞). In [13, Section 3], the function (6) was proved to be concave for x > 1. In [6, Theorem 1.1], it was proved that (n) n−1 ln Γ(x + 1) >0 (7) (−1) x ln x for x > 0 and n ∈ N. More strongly, the reciprocal of the function (6) was proved in [6, Theorem 1.4] to be a Stieltjes transform for x ∈ C\(−∞, 0], where C is the set of all complex numbers. Furthermore, among other things, it was directly shown in [10, Theorem 1.1] that the function (6) for x ∈ C \ (−∞, 0] is a Pick function. In [2, Lemma 4], it was demonstrated that the function ln Γ(x + 1) x ln(2x)

F (x) =

(8)

is strictly increasing on [1, ∞) and strictly concave on [46, ∞). With the help of this, the double inequality 1/(n ln n) a b Ωn exp ≤ 1/[(n+1) ln(n+1)] < exp (9) n(ln n)2 n(ln n)2 Ωn+1 was turned out in [2, Theorem 2] to be valid for n ≥ 2 if and only if 2(ln 2)2 ln(4π/3) 1 + ln(2π) = 0.3 . . . and b ≥ = 1.4 . . . . (10) 3 ln 3 2 It is clear that the function in (2) is equivalent to (8). Therefore, the above conjecture posed in [5, Remark 2.41] was verified once again. The first aim of this paper is to extend the ranges of x such that the function F (x) is both strictly increasing and strictly concave respectively as follows. 1 Theorem 1. On the interval 0, 2 , the function F (x) defined by (8) is strictly increasing; On the interval 12 , ∞ , it is both strictly increasing and strictly concave. a ≤ ln 2 ln π −

The second aim of this paper is, with the aid of Theorem 1, to generalize the 1/(n ln n) decreasing monotonicity of the sequence Ωn for n ≥ 2, the conjecture in [5, Remark 2.41], and the main result in [3, Corollary 3.1], to the logarithmic convexity. 1/(n ln n)

is strictly logarithmically convex for n ≥ 2.

Theorem 2. The sequence Ωn Consequently, the sequence

1/(n ln n)

Ωn

1/[(n+1) ln(n+1)]

Ωn+1 is strictly decreasing for n ≥ 2.

(11)

MONOTONICITY AND CONVEXITY RELATING TO VOLUME OF UNIT BALL

In [2, Lemma 3], the double inequality 2 ln x < 1− x ln x , G(x) < 1 3 ln(x + 1)

3

(12)

was verified to be true for x ≥ 3. The right-hand side inequality in (12) was also utilized in the proof of the inequality (9). The third aim of this paper is to extend and generalize the inequality (12) to a monotonicity result as follows. Theorem 3. The function G(x) defined in the inequality (12) is strictly increasing on (0, ∞) with lim+ G(x) = −∞ and lim G(x) = 1. (13) x→∞

x→0

2. Remarks Before proving our theorems, we are about to give some remarks on them and the volume of the unit ball in Rn . Remark 1. It is obvious that Theorem 1 extends or generalizes the conjecture posed in [5, Remark 2.41] and the corresponding conclusions obtained in [3, Corollary 3.1] and [2, Lemma 4] respectively. Remark 2. For a > 1 and x > 0 with x 6= a1 , let Fa (x) =

ln Γ(x + 1) . x ln(ax)

(14)

It is very natural to assert that the function Fa (x) is strictly increasing on 0, a1 and it is both strictly increasing and strictly concave on a1 , ∞ . More strongly, we conjecture that (−1)n−1 [Fa (x)](n) > 0 (15) for x >

1 a

and n ∈ N.

Remark 3. From Theorem 3, it is easy to see that the right-hand side inequality in (12) is sharp, but the left-hand side inequality can be sharpened by replacing 3) ln 3 the constant 23 = 0.666 · · · by a larger number 3(2 ln 2−ln = 0.683 . . . . 2 ln 2 Remark 4. We conjecture that (−1)k−1 [G(x)](k) > 0

(16)

for k ∈ N on (0, ∞), that is, the function 1 − G(x) is completely monotonic on (0, ∞). Remark 5. In [21] and its revised version [24], the reciprocal of the function [Γ(x + 1)]1/x was proved to be logarithmically completely monotonic on (−1, ∞). Consequently, the function 1/x √ π π x/2 = (17) Q(x) = Γ(1 + x/2) [Γ(1 + x/2)]1/x is also logarithmically completely monotonic on (−2, ∞). In particular, the function Q(x) is both strictly decreasing and strictly logarithmically convex on (−2, ∞). 1/n 1/n Because Q(n) = Ωn for n ∈ N, the sequence Ωn is strictly decreasing and

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B.-N. GUO AND F. QI

strictly logarithmically convex for n ∈ N. This generalizes one of the results in [5, Lemma 2.40] mentioned above. Furthermore, from the logarithmically complete monotonicity of Q(x), it is easy to obtain that the sequence 1/n Ωn (18) 1/(n+1) Ωn+1 is also strictly decreasing and strictly logarithmically convex for n ∈ N. As a direct consequence of the decreasing monotonicity of the sequence (18), the following double inequality may be derived: n 2 n/(n+1) n/(n+1) Ωn+1 , n ∈ N. (19) Ωn+1 < Ωn ≤ √ π When 1 ≤ n ≤ 4, the right-hand side inequality in (19) is better than the corresponding one in √ n/(n+1) 2 √ Ωn/(n+1) ≤ Ωn < e Ωn+1 , n∈N (20) π n+1 obtained in [1, Theorem 1]. Remark 6. In [15], the inequality [Γ(x + y + 1)/Γ(y + 1)]1/x < [Γ(x + y + 2)/Γ(y + 1)]1/(x+1)

r

x+y x+y+1

(21)

was confirmed to be valid if and only if x + y > y + 1 > 0 and to be reversed if and only if 0 < x + y < y + 1. Taking y = 0 and x = n2 in (21) leads to r 1/(n+2) Ωn+2 [Γ(n/2 + 1)]1/n n = < 4 , n > 2. (22) 1/(n+2) 1/n n+2 [Γ((n + 2)/2 + 1)] Ωn Similarly, if letting y = 1 and x = 1/(n+3)

Ωn+5

1/(n+1)

Ωn+3

n+1 2

> 1 in (21), then r n+3 1 , n ≥ 2. < 2/(n+1)(n+3) 4 n+5 π

(23)

Remark 7. In the preprint [23] and its formally published version [28], the following double inequality was discovered: For t > 0 and y > −1, the inequality a b x+y+1 [Γ(x + y + 1)/Γ(y + 1)]1/x x+y+1 < < (24) x+y+t+1 x+y+t+1 [Γ(x + y + t + 1)/Γ(y + 1)]1/(x+t) 1 1 holds with respect to x ∈ (−y − 1, ∞) if a ≥ max 1, y+1 and b ≤ min 1, 2(y+1) . Letting t = 1, y = 0 and x = n2 for n ∈ N in (24) reveals that r n+2 [Γ(n/2 + 1)]2/n n+2 < < n+4 n+4 [Γ((n + 2)/2 + 1)]2/(n+2) which is equivalent to r

1/(n+2)

Ω n+2 < n+2 1/n n+4 Ωn

r
1, we have x 1 x+1 1+ > . x [Γ(x + 1)]1/x

(30)

(31)

The inequality (31) is reversed for 0 < x < 1. Lemma 3 ([20, 26]). If t > 0, then t(2 + t) 2t < ln(1 + t) < ; 2+t 2(1 + t) If −1 < t < 0, the inequality (32) is reversed.

(32)

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B.-N. GUO AND F. QI

4. Proofs of theorems Proof of Theorem 1. The inequality (7) means that the function (6) is positive and increasing on (0, ∞). It is apparent that the function ln x 1 = ln(2x) 1 + ln 2/ ln x is positive and strictly increasing on 0, 12 and (1, ∞). Therefore, the function F (x) =

ln x ln Γ(x + 1) · x ln x ln(2x)

(33)

is strictly increasing on 0, 12 and (1, ∞). The inequality (31) for 0 < x < 1 in Lemma 2 can be rewritten as ln Γ(x + 1) < (1 − x) ln(x + 1) + x ln x. x A direct calculation yields F 0 (x) =

(34)

θ(x) x ln(2x)ψ(x + 1) − [ln(2x) + 1] ln Γ(x + 1) , 2 x2 [ln(2x)]2 x [ln(2x)]2

and ln Γ(x + 1) . x Utilizing the left-hand side inequality in (30) for k = 1, the inequality (34) and Lemma 3 leads to 1 1 θ0 (x) > x + ln(2x) − (1 − x) ln(x + 1) − x ln x x + 1 2(x + 1)2 1 1 2(2x − 1) x(1 − x)(2 + x) >x + − − x ln x x + 1 2(x + 1)2 1 + 2x 2(1 + x) 4 2x + 5x3 + 8x2 + 3x − 8 =x − ln x 2(x + 1)2 (2x + 1) 4 2x + 5x3 + 8x2 + 3x − 8 2(x − 1) >x − 2(x + 1)2 (2x + 1) 1+x 4 3 2 x 2x − 3x + 4x + 11x − 4 = 2(x + 1)2 (2x + 1) 2 x[2x (x − 1)2 + x(x + 1)2 + 10x − 4] = 2(x + 1)2 (2x + 1) >0 for 21 < x < 1. Hence, the function θ(x) is strictly increasing on 12 , 1 . Since 1 3 1 θ = − ln Γ =− ln π − ln 2 > 0, 2 2 2 θ0 (x) = x ln(2x)ψ 0 (x + 1) −

the function θ(x), and so the function F 0 (x), is positive, and then the function F (x) 1 1 is increasing on 2 , 1 . In a word, the function F (x) is strictly increasing on 0, 2 1 and 2 , ∞ . A ready computation gives


7

x3 [ln(2x)]3 F 00 (x) = ln Γ(x + 1) 2[ln(2x)]2 + 3 ln(2x) + 2 x2 [ln(2x)]2 ψ 0 (x + 1) − 2x ln(2x)[ln(2x) + 1]ψ(x + 1) + 2[ln(2x)]2 + 3 ln(2x) + 2 and

d x3 [ln(2x)]3 F 00 (x) [ln(2x)]2 φ(x) = 2 2 d x 2[ln(2x)] + 3 ln(2x) + 2 2[ln(2x)]2 + 3 ln(2x) + 2

for x > 21 , where, by making use of the right-hand side inequality in (29) and the inequality (30) for k = 1, 2, φ(x) = [2 ln(2x) + 5]ψ(x + 1) + x x 2[ln(2x)]2 + 3 ln(2x) + 2 ψ 00 (x + 1) − [4 ln(2x) + 3]ψ 0 (x + 1) 1 < [2 ln(2x) + 5] ln(x + 1) − 2(x + 1) 1 1 2 + x −x 2[ln(2x)] + 3 ln(2x) + 2 + (x + 1)2 (x + 1)3 1 1 − [4 ln(2x) + 3] + x + 1 2(x + 1)2 1 = [2 ln(2x) + 5] ln(x + 1) − 2(x + 1) 2 x 4x(x + 2)[ln(2x)] + 2 7x2 + 16x + 6 ln(2x) + 10x2 + 23x + 9 − 2(x + 1)3 , ϕ(x) and −x(x + 1)4 ϕ0 (x) = 2x4 + 10x3 + 19x2 + 6x + 1 + 2(x + 4)x2 [ln(2x)]2 + x 2x3 + 10x2 + 20x + 3 ln(2x) − 2(x + 1)4 ln(x + 1) , h(x), h0 (x) = 8x3 + 34x2 + 52x + 7 + 2x(3x + 8)[ln(2x)]2 + 8x3 + 34x2 + 56x + 3 ln(2x) − 8(x + 1)3 ln(x + 1), xh00 (x) = 24x3 + 86x2 + 100x + 3 + 4x(3x + 4)[ln(2x)]2 + 8x 3x2 + 10x + 11 ln(2x) − 24x(x + 1)2 ln(x + 1) , q(x), q 0 (x) = 4 18x2 + 57x + 47 + (6x + 4)[ln(2x)]2 + 2 9x2 + 23x + 15 ln(2x) − 6 3x2 + 4x + 1 ln(x + 1) , xq 00 (x) = 4 36x2 + 97x + 30 + 6[ln(2x)]2 x − 12(3x + 2) ln(x + 1)x + 36x2 + 58x + 8 ln(2x) , x2 (x + 1)q (3) (x) = 8 18x3 + 53x2 + 18x − 11 − 18(x + 1)x2 ln(x + 1) + 2 9x3 + 12x2 + x − 2 ln(2x) , 8p(x),

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B.-N. GUO AND F. QI

xp0 (x) = 2 27x3 + 65x2 + 10x − 2 − 9(3x + 2)x2 ln(x + 1) + 27x2 + 24x + 1 x ln(2x) , (x + 1)x2 p00 (x) = 2 54x4 + 152x3 + 90x2 + 3x + 2 + 6 9x2 + 13x + 4 x2 ln(2x) − 18 3x2 + 4x + 1 x2 ln(x + 1) , (x + 1)2 x3 p(3) (x) = 2 54x5 + 168x4 + 146x3 + 18x2 − 9x − 4 + 54(x + 1)2 x3 ln(2x) − 54(x + 1)2 x3 ln(x + 1) , (x + 1)3 x4 p(4) (x) = −4 3x5 + 8x4 − 9x2 − 19x − 6 , −4λ(x), 0

λ (x) = 15x4 + 32x3 − 18x − 19, λ00 (x) = 60x3 + 96x2 − 18. 00 Since λ00 (x) is strictly increasing for x > 21 and λ00 12 = 27 2 , the function λ (x) > 0, 1 369 0 0 1 so λ (x) is strictly increasing for x > 2 . From λ 2 = − 16 and limx→∞ λ0 (x) = ∞, it follows that the function λ0 (x) has a unique zero which is a minimum of 1 1 λ(x) for x > 2 . Since λ 2 = − 549 32 and limx→∞ λ(x) = ∞, the functions λ(x) and p(4) (x) has a unique zero which is a maximum of p(3) (x). Due to p(3) 21 = 188 − 108 ln 32 = 144.20 · · · and limx→∞ p(3) (x) = 108(1 + ln 2), the function 00 p(3) (x) is for x > 12 . Owing positive, thus 3the function p (x) is strictly increasing 00 1 00 to p 2 = 258 − 90 ln 2 = 221.50 . . . , the function p (x) is positive and p0 (x) is strictly increasing for x > 12 . By virtue of p0 21 = 12 181 − 63 ln 32 = 77.72 . . . , it is easy to see that p0 (x) > 0 and p(x) is strictly increasing for x > 21 . Owing 3 to p 12 = 27 = 10.76 . . . , it is deduced that p(x) > 0 and q (3) (x) > 0 4 2 − ln 2 1 for x > 2 , consequently the function q 00 (x) is strictly increasing for x > 21 . On account of q 00 12 = 700 − 168 ln 32 = 631.88 . . . , we obtain that q 00 (x) > 0 and q 0 (x) is strictly increasing for x > 12 . In virtue of q 0 21 = 320 − 90 ln 32 , we have 1 q 0 (x) and then the > 0, function q(x) is strictly increasing for x > 2 00. Because of 155 1 3 q 2 = 2 − 27 ln 2 = 66.55 . . . , we acquire that q(x) > 0 and h (x) > 0, so 3 h0 (x) is strictly increasing for x > 12 . By h0 12 = 85 − 27 ln 2 2 = 33.55 . . ., it 1 1 0 is derived that h (x) > 0 and h(x) is strictly increasing0 for x > 2 . Since h 2 = 81 3 = 6.01 . . . , we gain that h(x) > 8 1 − ln 2 0 and ϕ (x) 21 . Due to ϕ 12 = − 91 + 5 ln 27 2 = −1.34 . . . , it is procured that ϕ(x) < 0, i.e., φ(x) < 0, hence d x3 [ln(2x)]3 F 00 (x) 21 . By √ x3 [ln(2x)]3 F 00 (x) π = ln = −0.12 . . . , 2 2[ln(2x)] + 3 ln(2x) + 2 x=1/2 2


it follows that

9

x3 [ln(2x)]3 F 00 (x) < 0, 2[ln(2x)]2 + 3 ln(2x) + 2

1 00 which is equivalent to F (x) < 0 for x > 2 . As a result, the function F (x) is 1 concave on 2 , ∞ . The proof of Theorem 1 is complete.

Proof of Theorem 2. Let f (x) =

π x/2 Γ(1 + x/2)

1/(x ln x) (35)

for x > 0. Taking the logarithm of f (x) gives ln π ln Γ(1 + x/2) ln π x ln f (x) = − = − 2F , 2 ln x x ln x 2 ln x 2 where F (x) is defined by (8) for x > 0 and x 6= 12 . Differentiating twice gives (ln x + 2) ln π 1 00 x 00 [ln f (x)] = − F . 2x2 (ln x)3 2 2 Since F (x) is strictly concave for x > 12 , then the function [ln f (x)]00 is positive for x > 1. As a result, the function f (x) is strictly logarithmically convex for x > 1, 1/(n ln n) for n ≥ 2 is strictly logarithmically convex. and so the sequence f (n) = Ωn Since the function f (x) is strictly logarithmically convex for x > 1, then [ln f (x)]0 is strictly increasing for x > 1, therefore 0 f (x) ln = [ln f (x) − ln f (x + 1)]0 = [ln f (x)]0 − [ln f (x + 1)]0 < 0 f (x + 1) for x > 1. This implies that the function

f (x) f (x+1)

is strictly decreasing for x > 1,

f (n) f (n+1)

hence the sequence is also strictly decreasing for n ≥ 2. The proof of Theorem 2 is thus completed. Proof of Theorem 3. It is easy to see that the function G(x) in (12) may be rearranged as x 1 ln x ln 1 + . (36) G(x) = ln(x + 1) x x It is common knowledge that the function 1+ x1 is strictly increasing and greater than 1 on (0, ∞) and tends to e as x → ∞. Furthermore, for x > 0, 0 ln x (x + 1) ln(x + 1) − x ln x = > 0. ln(x + 1) x(x + 1)[ln(x + 1)]2 ln x Thus, the function ln(x+1) is strictly increasing on (0, ∞) and positive on (1, ∞) and, by L’Hˆ ospital’s rule, tends to 1 as x → ∞. Therefore, the second limit in (13) is valid and the function G(x) is strictly increasing on (1, ∞). The first limit in (13) can be calculated by L’Hˆ ospital’s rule as follows:

1 − ln x/ln(x + 1) lim ln x 1/x x→0+ x[(x + 1) ln(x + 1) − x ln x] lim ln x = lim+ (x + 1)[ln(x + 1)]2 x→0+ x→0

lim G(x) = lim+

x→0+

x→0

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B.-N. GUO AND F. QI

x x ln x 1− lim ln x ln(x + 1) (x + 1) ln(x + 1) x→0 x→0+ ln x x = lim 1 − · lim ln x ln(x + 1) x + 1 x→0+ x→0+ = −∞. = lim+

The function G(x) can also be rearranged as x ln x G(x) = [ln(x + 1) − ln x] , f1 (x)f2 (x) ln(x + 1)

(37)

for x ∈ (0, ∞). It is not difficult to see that the function f2 (x) is positive and decreasing on (0, ∞). Straightforward differentiation produces f10 (x) =

g(x) ln(x + 1) + [ln(x + 1) − x/(x + 1)] ln x , 2 [ln(x + 1)] [ln(x + 1)]2

and f1 (1) = 0. By virtue of the double inequality (32), it follows that 2x x(2 + x) x g(x) > + − ln x 2+x 2(1 + x) x + 1 2x x2 = + ln x 2 + x 2(1 + x) 4(1 + x) x2 + ln x = 2(1 + x) x(2 + x) x2 , h(x) 2(1 + x) and x3 − 4x − 8 0 on (0, 1). This means that the functions g(x) and f10 (x) are positive on (0, 1). Hence, the function f1 (x) is increasing and negative on (0, 1). In conclusion, the function G(x) = f1 (x)f2 (x) is increasing and negative on (0, 1). The proof of Theorem 3 is complete. Second proof of a part of Theorem 3. By (37), it is clear that the function Z x+1 Z 1 1 1 f2 (x) = dt = dt t t + x x 0 is completely monotonic on (0, ∞). On the other hand, we have f1 (x) = and

u(x) − u(0) v(x) − v(0)

u0 (x) = (x + 1)(1 + ln x) v 0 (x) is increasing on (0, ∞), where ( x ln x, x ∈ (0, 1] u(x) = 0, x=0


11

and v(x) = ln(x + 1) for x ∈ [0, 1]. The monotonic form of L’Hˆospital’s rule put forward in [4, Theorem 1.25] (or see [19, p. 92, Lemma 1] and [27, p. 10, Lemma 2.9]) reads that if U and V are continuous on [a, b] and differentiable on 0 (a, b) such that V 0 (x) 6= 0 and VU 0 (x) (x) is increasing (or decreasing) on (a, b), then (x)−U (b) U (x)−U (a) the functions VU (x)−V (b) and V (x)−V (a) are also increasing (or decreasing) on (a, b). Therefore, the function f1 (x) is increasing and negative on (0, 1). In a word, the function G(x) = f1 (x)f2 (x) is increasing and negative on (0, 1).

5. Addenda Recently some new results seemingly motivated by the preprint [22] of this paper have been established. For completeness, keeping academic clues, and basing on comments of referees, we would like to add these new developments in the form of addenda as follows. 5.1. The conjecture posed in our Remark 2 has been proved in [9] as a consequence of an integral representation for the function Fa (x) defined by (14). These results are stronger than our Theorem 1, but they refer to the preprint [22] of this paper, and it seems that the preprint [22] has been an inspiration for [9]. 5.2. We also mention that the sequence an+2 , defined by (4), is proved in [9] to be a Hausdorff moment sequence and this result is stronger than our Theorem 2. 5.3. The conjecture posed in Remark 4 has been proved in the preprint [7] and its formally published version [8]. 5.4. In the paper [18], important ideas for the volume Ωn of n-dimensional unit ball Bn are included. So we suggest readers having interests in this area to carefully read it. Acknowledgements. The authors heartily appreciate referees for their valuable comments and pointing out the recently appeared references [7, 9, 18]. References [1] H. Alzer, Inequalities for the volume of the unit ball in Rn , J. Math. Anal. Appl. 252 (2000), 353–363. [2] H. Alzer, Inequalities for the volume of the unit ball in Rn , II, Mediterr. J. Math. 5 (2008), no. 4, 395–413. [3] G. D. Anderson and S.-L. Qiu, A monotoneity property of the gamma function, Proc. Amer. Math. Soc. 125 (1997), 3355–3362. [4] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Conformal Invariants, Inequalities, and Quasi-conformal Maps, John Wiley & Sons, New York, 1997. [5] G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, Special functions of quasiconformal theory, Expo. Math. 7 (1989), 97–136. [6] C. Berg and H. L. Pedersen, A completely monotone function related to the gamma function, J. Comput. Appl. Math. 133 (2001), no. 1-2, 219–230. [7] C. Berg and H. L. Pedersen, A completely monotonic function used in an inequality of Alzer, arXiv preprint (2011), available online at http://arxiv.org/abs/1105.6181. [8] C. Berg and H. L. Pedersen, A completely monotonic function used in an inequality of Alzer, Comput. Methods Funct. Theory 12 (2012), no. 1, 329–341. [9] C. Berg and H. L. Pedersen, A one-parameter family of Pick functions defined by the Gamma function and related to the volume of the unit ball in n-space, Proc. Amer. Math. Soc. 139 (2011), no. 6, 2121–2132; Available online at http://dx.doi.org/10.1090/ S0002-9939-2010-10636-6.

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[10] C. Berg and H. L. Pedersen, Pick functions related to the gamma function, Conference on Special Functions (Tempe, AZ, 2000). Rocky Mountain J. Math. 32 (2002), no. 2, 507–525. [11] C.-P. Chen and F. Qi, Inequalities relating to the gamma function, Aust. J. Math. Anal. Appl. 1 (2004), no. 1, Art. 3; Available online at http://ajmaa.org/cgi-bin/paper.pl? string=v1n1/V1I1P3.tex. [12] C.-P. Chen and F. Qi, Note on a monotonicity property of the gamma function, Octogon Math. Mag. 12 (2004), no. 1, 123–125. ´ Elbert and A. Laforgia, On some properties of the gamma function, Proc. Amer. Math. [13] A. Soc. 128 (2000), no. 9, 2667–2673. [14] B.-N. Guo, R.-J. Chen, and F. Qi, A class of completely monotonic functions involving the polygamma functions, J. Math. Anal. Approx. Theory 1 (2006), no. 2, 124–134. [15] B.-N. Guo and F. Qi, An extension of an inequality for ratios of gamma functions, J. Approx. Theory 163 (2011), no. 9, 1208–1216; Available online at http://dx.doi.org/10.1016/j. jat.2011.04.003. [16] B.-N. Guo and F. Qi, Two new proofs of the complete monotonicity of a function involving the psi function, Bull. Korean Math. Soc. 47 (2010), no. 1, 103–111; Available online at http://dx.doi.org/10.4134/bkms.2010.47.1.103. [17] X. Li, Monotonicity properties for the gamma and psi functions, Sci. Magna 4 (2008), no. 4, 18–23. [18] C. Mortici, Monotonicity properties of the volume of the unit ball in Rn , Optim. Lett. 4 (2010), no. 3, 457–464. [19] D.-W. Niu, J. Cao, and F. Qi, Generalizations of Jordan’s inequality and concerned relations, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 72 (2010), no. 3, 85–98. [20] F. Qi and C.-P. Chen, Monotonicities of two sequences, Mathematics and Informatics Quarterly 9 (1999), no. 4, 136–139. [21] F. Qi and B.-N. Guo, Complete monotonicities of functions involving the gamma and digamma functions, RGMIA Res. Rep. Coll. 7 (2004), no. 1, Art. 8, 63–72; Available online at http://rgmia.org/v7n1.php. [22] F. Qi and B.-N. Guo, Monotonicity and logarithmic convexity relating to the volume of the unit ball, arXiv preprint (2009), available online at http://arxiv.org/abs/0902.2509. [23] F. Qi and B.-N. Guo, Necessary and sufficient conditions for a function involving a ratio of gamma functions to be logarithmically completely monotonic, arXiv preprint (2009), available online at http://arxiv.org/abs/0904.1101. [24] F. Qi and B.-N. Guo, Some logarithmically completely monotonic functions related to the gamma function, J. Korean Math. Soc. 47 (2010), no. 6, 1283–1297; Available online at http://dx.doi.org/10.4134/JKMS.2010.47.6.1283. [25] F. Qi, S. Guo, and B.-N. Guo, Complete monotonicity of some functions involving polygamma functions, J. Comput. Appl. Math. 233 (2010), no. 9, 2149–2160; Available online at http: //dx.doi.org/10.1016/j.cam.2009.09.044. [26] F. Qi, W. Li, and B.-N. Guo, Generalizations of a theorem of I. Schur, Appl. Math. E-Notes 6 (2006), 244–250. [27] F. Qi, D.-W. Niu, and B.-N. Guo, Refinements, generalizations, and applications of Jordan’s inequality and related problems, J. Inequal. Appl. 2009 (2009), Article ID 271923, 52 pages; Available online at http://dx.doi.org/10.1155/2009/271923. [28] F. Qi, C.-F. Wei, and B.-N. Guo, Complete monotonicity of a function involving the ratio of gamma functions and applications, Banach J. Math. Anal. 6 (2012), no. 1, 35–44; Available online at https://doi.org/10.15352/bjma/1337014663. (Guo) School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China Email address: [email protected], [email protected] (Qi) School of Mathematics and Informatics, Henan Polytechnic University, Jiaozuo City, Henan Province, 454010, China Email address: [email protected], [email protected], [email protected] URL: https://qifeng618.wordpress.com