JFS Journal of Function Spaces 2314-8888 2314-8896 Hindawi Publishing Corporation 10.1155/2016/4165601 4165601 Research Article A Sharp Lower Bound for Toader-Qi Mean with Applications Yang Zhen-Hang Chu Yu-Ming 1 Zhu Kehe School of Mathematics and Computation Science Hunan City University Yiyang 413000 China hncu.net 2016 17 1 2016 2016 28 10 2015 10 12 2015 24 12 2015 2016 Copyright © 2016 Zhen-Hang Yang and Yu-Ming Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We prove that the inequality T Q ( a , b ) > L p ( a , b ) holds for all a , b > 0 with a b if and only if p 3 / 2 , where T Q ( a , b ) = 2 / π 0 π / 2 a c o s 2 θ b s i n 2 θ d θ , L p ( a , b ) = [ ( b p - a p ) / ( p ( b - a ) ) ] 1 / p ( p 0 ) , and L 0 ( a , b ) = a b 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.

1. Introduction

Let p R and a , b > 0 with a b . Then the Toader-Qi mean T Q ( a , b )  and p -order logarithmic mean L p ( a , b ) are defined by (1) T Q a , b = 2 π 0 π / 2 a c o s 2 θ b s i n 2 θ d θ , (2) L p a , b = b p - a p p log b - log a 1 / p p 0 , L 0 a , b = l i m p 0 L p a , b = a b , respectively. In particular, L 1 ( a , b ) = L ( a , b ) is the classical logarithmic mean of a and b .

It is well-known that the p -order logarithmic mean L p ( a , b ) is continuous and strictly increasing with respect to p R for fixed a , b > 0 with a b . Recently, the Toader-Qi and p -order logarithmic means have been the subject of intensive research. In particular, many remarkable inequalities for the Toader-Qi and p -order logarithmic means can be found in the literature .

In , Qi et al. proved that the identity (3) T Q a , b = a b I 0 1 2 log b a and the inequalities (4) L a , b < T Q a , b < A a , b + G a , b 2 < 2 A a , b + G a , b 3 < I a , b hold for all a , b > 0 with a b , where (5) I 0 t = n = 0 t 2 n 2 2 n n ! 2 is the modified Bessel function of the first kind  and A ( a , b ) = ( a + b ) / 2 , G ( a , b ) = a b and I ( a , b ) = ( b b / a a ) 1 / ( b - a ) / e are, respectively, the classical arithmetic, geometric, and identric means of a and b .

In , Yang proved that the double inequalities (6) 2 A a , b L a , b π < T Q a , b < A a , b L a , b , (7) A 1 / 4 a , b L 3 / 4 a , b < T Q a , b < 1 4 A a , b + 3 4 L a , b and conjectured that the inequalities (8) T Q a , b < I 1 / 2 a , b L 1 / 2 a , b , (9) T Q a , b > L 3 / 2 a , b hold for all a , b > 0 with a b . Inequality (8) was proved by Yang et al. in .

Let b > a > 0 and t = ( log b - log a ) / 2 > 0 . Then from (1)–(3) we clearly see that (10) L p a , b = a b sinh p t p t 1 / p p 0 , T Q a , b = 2 a b π 0 π / 2 e t c o s 2 θ d θ = a b I 0 t = 2 a b π 0 π / 2 cosh t cos θ d θ = 2 a b π 0 π / 2 cosh t sin θ d θ .

The main purpose of this paper is to give a positive answer to the conjecture given by (9). As applications, we present two fine inequalities chains for certain bivariate means and a lower bound for the kernel function of the Szász-Mirakjan-Durrmeyer operator.

2. Lemmas

In order to prove our main result we need several lemmas, which we present in this section.

Lemma 1 (see [<xref ref-type="bibr" rid="B10">10</xref>]).

The double inequality (11) 1 x + a 1 - a < Γ x + a Γ x + 1 < 1 x 1 - a holds for all x > 0 and a ( 0,1 ) , where Γ ( x ) = 0 e - t t x - 1 d t is the classical Euler gamma function.

Lemma 2 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

Let I 0 ( t ) be defined by (5). Then the identity (12) I 0 2 t = n = 0 2 n ! 2 2 n n ! 4 t 2 n holds for all t R .

Lemma 3 (see [<xref ref-type="bibr" rid="B3">3</xref>]).

The Wallis ratio (13) W n = 2 n - 1 ! ! 2 n ! ! = 2 n ! 2 2 n n ! 2 = Γ n + 1 / 2 Γ 1 / 2 Γ n + 1 is strictly decreasing and log-convex with respect to all integers n 0 .

Lemma 4.

The identity (14) k = 0 n a 2 k 2 k ! 2 n - 2 k ! = a + 1 2 n + a - 1 2 n 2 2 n ! holds for all a R and n N .

Proof.

Let n k = n ! / k ! ( n - k ) ! be the number of combinations of n objects taken k at a time. Then from the well-known binomial theorem we have (15) a + 1 2 n = k = 0 2 n 2 n k a k = k = 0 n 2 n 2 k a 2 k + k = 1 n 2 n 2 k - 1 a 2 k - 1 , a - 1 2 n = k = 0 2 n 2 n k - 1 2 n - k a k = k = 0 n 2 n 2 k a 2 k - k = 1 n 2 n 2 k - 1 a 2 k - 1 .

Equation (15) leads to (16) a + 1 2 n + a - 1 2 n 2 = k = 0 n 2 n 2 k a 2 k = k = 0 n 2 n ! a 2 k 2 k ! 2 n - 2 k ! .

Lemma 5.

Let k , n N with k n and (17) u k , n = 2 k ! 2 2 n k ! 4 n - k ! 2 . Then (18) u k , n > 2 2 π π n + 1 2 n + 1 2 2 k 2 k ! 2 n - 2 k ! for all n 8 .

Proof.

Let W n be defined by (13). Then it follows from Lemmas 1 and 3 together with (17) and Γ ( 1 / 2 ) = π that (19) u k , n = W k 2 W n - k 2 2 k 2 k ! 2 n - 2 k ! W k W n / 2 2 2 2 k 2 k ! 2 n - 2 k ! W n W n / 2 2 2 2 k 2 k ! 2 n - 2 k ! = 1 π π Γ n + 1 / 2 Γ n + 1 Γ n / 2 + 1 / 2 Γ n / 2 + 1 2 2 2 k 2 k ! 2 n - 2 k ! > 1 π π 1 n + 1 / 2 1 n / 2 + 1 / 2 2 2 2 k 2 k ! 2 n - 2 k ! = 2 2 π π n + 1 2 n + 1 2 2 k 2 k ! 2 n - 2 k ! for all n 8 and 0 k n .

3. Main Result Theorem 6.

The inequality (20) T Q a , b > L p a , b holds for all a , b > 0 with a b if and only if p 3 / 2 .

Proof.

Since both the Toader-Qi mean T Q ( a , b ) and p -order logarithmic mean L p ( a , b ) are symmetric and homogeneous and T Q ( a , b ) > L ( a , b ) and L p ( a , b ) is strictly increasing with respect to p R for all a , b > 0 with a b , without loss of generality, we assume that p > 1 and b > a > 0 . Let t = ( log b - log a ) / 2 > 0 . Then it follows from (10) that inequality (20) is equivalent to (21) I 0 t > sinh p t p t 1 / p for all t > 0 .

If inequality (21) holds for all t > 0 . Then (5) and (21) lead to (22) l i m t 0 + I 0 t - sinh p t / p t 1 / p t 2 = - 1 6 p - 3 2 0 , which gives p 3 / 2 .

Next, we only need to prove that inequality (21) holds for p = 3 / 2 and all t > 0 ; that is (23) I 0 3 t > sinh 3 t / 2 3 t / 2 2 .

It follows from (5) and Lemma 2 that (24) I 0 3 t = n = 0 t 2 n 2 2 n n ! 2 3 = n = 0 2 n ! t 2 n 2 2 n n ! 4 n = 0 t 2 n 2 2 n n ! 2 = n = 0 k = 0 n 2 k ! 2 2 k k ! 4 1 2 2 n - k n - k ! 2 t 2 n = n = 0 k = 0 n u k , n t 2 n , where u k , n is defined as in (17).

Note that (25) sinh 3 t / 2 3 t / 2 2 = 2 cosh 3 t - 1 9 t 2 = 2 n = 0 3 2 n t 2 n 2 n + 2 ! .

Let (26) v n = k = 0 n u k , n - 2 × 3 2 n 2 n + 2 ! .

Then simple computations lead to (27) v 0 = v 1 = 0 , v 2 = 3 320 , v 3 = 113 26880 , v 4 = 2057 2867200 , v 5 = 1741 25231360 , v 6 = 4335377 991895224320 , v 7 = 2186227 11021058048000 .

From Lemmas 4 and 5 together with (24)–(26), we have (28) I 0 3 t - sinh 3 t / 2 3 t / 2 2 = n = 0 v n t 2 n , v n > 2 2 π π n + 1 2 n + 1 3 2 n + 1 2 2 n ! - 2 × 3 2 n 2 n + 2 ! = 2 2 n + 1 - π π 3 2 n + 2 2 n + 1 π π n + 1 2 n + 1 ! > 0 for all n 8 .

Therefore, inequality (23) follows from (27) and (28).

Remark 7.

Theorem 6 gives a positive answer to the conjecture given by (9).

Remark 8.

It follows from (23) that the inequality (29) I 0 3 t > 2 cosh 3 t - 1 9 t 2 holds for all t 0 .

4. Applications

For a , b > 0 , the Toader mean T ( a , b )  and arithmetic-geometric mean A G M ( a , b )  are, respectively, defined by (30) T a , b = 2 π 0 π / 2 a 2 cos 2 θ + b 2 sin 2 θ d θ , A G M a , b = l i m n a n = l i m n b n , where a n and b n are given by (31) a 0 = a , b 0 = b , a n + 1 = a n + b n 2 = A a n , b n , b n + 1 = a n b n = G a n , b n .

Let T p ( a , b ) = T 1 / p a p , b p and I q ( a , b ) = I 1 / q a q , b q be the p -order Toader and q -order identric means of a and b , respectively. Then Theorem 6 leads to two fine inequalities chains for certain bivariate means.

Theorem 9.

The inequalities (32) L a , b < A G M a , b < A 1 / 4 a , b L 3 / 4 a , b < L 3 / 2 a , b < T Q a , b < 1 4 A a , b + 3 4 L a , b < 1 2 L a , b + 1 2 I a , b < 1 2 A a , b + 1 2 G a , b < T 1 / 3 a , b < I 3 / 4 a , b , L a , b < A G M a , b < A 1 / 4 a , b L 3 / 4 a , b < L 3 / 2 a , b < T Q a , b < L 1 / 2 a , b I 1 / 2 a , b < 1 2 L a , b + 1 2 I a , b < 1 2 A a , b + 1 2 G a , b < T 1 / 3 a , b < I 3 / 4 a , b hold for all a , b > 0 with a b .

Proof.

The following inequalities can be found in the literature [3, 4, 7, 1214]: (33) A a , b + G a , b 2 < T 1 / 3 a , b < I 3 / 4 a , b , (34) L a , b < A G M a , b < L 3 / 4 a , b A 1 / 4 a , b < L 3 / 2 a , b , (35) I a , b > L a , b + A a , b 2 , L a , b + I a , b < A a , b + G a , b for all a , b > 0 with a b .

It follows from (35) that (36) A a , b + G a , b 2 > I a , b + L a , b 2 > 3 4 L a , b + 1 4 A a , b for all a , b > 0 with a b .

Therefore, inequality (32) follows easily from (7), (8), (33), (34), (36), and Theorem 6.

Remark 10.

Let b > a > 0 and t = ( log b - log a ) / 2 > 0 . Then simple computations lead to (37) L a , b a b = sinh t t , I a , b a b = e t cosh t / sinh t - 1 , A a , b a b = cosh t .

Note that (38) lim t 0 + sinh t / t e t cosh t / sinh t - 1 - 3 sinh t / 4 t + cosh t / 4 2 t 4 = 1 720 , lim t sinh t t e t cosh t / sinh t - 1 - 3 sinh t 4 t + cosh t 4 2 = - .

Inequalities (37) and (38) imply that there exist small enough δ > 0 and large enough M > 1 such that (39) I 1 / 2 a , b L 1 / 2 a , b > 1 4 A a , b + 3 4 L a , b for all b > a > 0 with b / a ( 1,1 + δ ) and (40) I 1 / 2 a , b L 1 / 2 a , b < 1 4 A a , b + 3 4 L a , b for all b > a > 0 with b / a ( M , ) .

Let x [ 0 , ) , n > 0 , k 0 , p n , k ( x ) = ( n x ) k e - n x / k ! , and f L p ( [ 0 , ) ) ( 1 p ) . Then the kernel function T n ( x , y ) of the Szász-Mirakjan-Durrmeyer operator  (41) M n f ; x = k = 0 f , p n , k 1 , p n , k p n , k x = n f , T n x , · 0 T n x , y f y d y is given by (42) T n x , y = k = 0 p n , k x p n , k y = e - n x + y I 0 2 n x y .

Berdysheva  proved that T n ( x , y ) is completely monotonic with respect to n > 0 for fixed x , y [ 0 , ) and (43) T n x , y e - n x - y 2 for all x , y [ 0 , ) .

From Remark 8 and (42), we get a lower bound for the kernel function T n ( x , y ) immediately.

Corollary 11.

The inequality (44) T n x , y > e - n x + y cosh 6 n x y - 1 18 n 2 x y 1 / 3 holds for all x , y ( 0 , ) .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research was supported by the Natural Science Foundation of China under Grant 61374086 and the Natural Science Foundation of Zhejiang Province under Grant LY13A010004.

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