Yang and Chu Journal of Inequalities and Applications (2017) 2017:41 DOI 10.1186/s13660-017-1317-z
RESEARCH
Open Access
On approximating the modified Bessel function of the second kind Zhen-Hang Yang1,2 and Yu-Ming Chu1* *
Correspondence: [email protected] 1 School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article
Abstract In the article, we prove that the double inequalities √ –x √ –x πe πe < K0 (x) < √ , √ 2(x + a) 2(x + b)
1+
1 K1 (x) 1 < 0 if and only if a ≥ 1/4 and b = 0 if a, b ∈ [0, ∞), where Kν (x) is the modified Bessel function of the second kind. As applications, we provide bounds for Kn+1 (x)/Kn (x) with n ∈ N and present the necessary and sufficient condition such that √ the function x → x + pex K0 (x) is strictly increasing (decreasing) on (0, ∞). MSC: 33B10; 26A48 Keywords: modified Bessel function; gamma function; monotonicity
1 Introduction The modified Bessel function of the first kind Iν (x) is a particular solution of the secondorder differential equation x y (x) + xy (x) – x + ν y(x) = , and it can be expressed by the infinite series Iν (x) =
∞ n=
n+ν x . n!(ν + n + )
While the modified Bessel function of the second kind Kν (x) is defined by Kν (x) =
π(I–ν (x) – Iν (x)) , sin(πν)
(.)
where the right-hand side of the identity of (.) is the limiting value in case ν is an integer. The following integral representation formula and asymptotic formulas for the modified Bessel function of the second kind Kν (x) can be found in the literature [], .., .., .., ..:
∞
Kν (x) =
e–x cosh(t) cosh(νt) dt
(x > ),
(.)
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
K (x) ∼ – log x (x → ), –ν x (ν > , x → ), Kν (x) ∼ (ν)
ν – (ν – )(ν – ) π –x e + + + · · · Kν (x) ∼ x x !(x)
Page 2 of 8
(.) (.) (x → ∞).
(.)
From (.) we clearly see that
∞
K (x) =
–x cosh(t)
e
K (x) = –
∞
dt =
∞
e–xt dt, √ t –
te–xt dt = –K (x). √ t –
(.) (.)
Recently, the bounds for the modified Bessel function of the second kind Kν (x) have attracted the attention of many researchers. Luke [] proved that the double inequality √ x x + x < e K (x) < √ x + π (x + ) x
(.)
holds for all x > . Gaunt [] proved that the double inequality
x+
(x + ) < < (x + )
x e K (x) < √ π x
(.)
∞ takes place for all x > , where (x) = t x– e–t dx is the classical gamma function. In [], Segura proved that the double inequality ν+
√
x
x
+ ν
and ν ≥ . Bordelon and Ross [] and Paris [] provided the inequality ν Kν (x) y–x x >e Kν (y) y
(.)
for all ν > –/ and y > x > . Laforgia [] established the inequality –ν Kν (x) y–x x >e Kν (y) y for all y > x > if ν ∈ (, /), and inequality (.) is reversed if ν ∈ (/, ∞). Baricz [] presented the inequality –/ Kν (x) y–x x >e Kν (y) y for all y > x > and ν ∈ (–∞, –/) ∪ (/, ∞).
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 3 of 8
Motivated by inequality (.), in the article, we prove that the double inequality √ –x √ –x πe πe < K (x) < √ √ (x + a) (x + b) holds for all x > if and only if a ≥ / and b = if a, b ∈ [, ∞). As applications, we provide bounds for Kn+ (x)/Kn (x) with n ∈ N and present the necessary and sufficient condition √ such that the function x → x + pex K (x) is strictly increasing (decreasing) on (, ∞).
2 Lemmas In order to prove our main results, we need two lemmas which we present in this section. Lemma . (See []) Let –∞ ≤ a < b ≤ ∞, f , g : [a, b] → R be continuous on [a, b] and differentiable on (a, b), and g (x) = on (a, b). If f (x)/g (x) is increasing (decreasing) on (a, b), then so are the functions f (x) – f (b) . g(x) – g(b)
f (x) – f (a) , g(x) – g(a)
If f (x)/g (x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma . The function x → f (x) =
K (x) –x [K (x) – K (x)]
(.)
is strictly increasing from (, ∞) onto (, /). Proof Let ω(t) =
√
(t – )/(t + ). Then it follows from (.), (.) and (.) that
∞
K (x) – K (x) =
x K (x) – K (x) = –
ω(t)e–xt dt,
∞
ω(t)d e–xt
= ω(t)e–xt |t= t=∞
∞
+
ω (t)e–xt dt
∞
t– e–xt dt, – )/ ∞
ω(t) –xt K (x) – x K (x) – K (x) = e dt, t + ∞ ω(t) –xt e dt K (x) – x[K (x) – K (x)] = ∞ t+ –xt , f (x) = [K (x) – K (x)] ω(t)e dt ∞ tω(t) –xt ∞ ∞ ∞ – t+ e dt ω(t)e–xt dt + ω(t) e–xt dt tω(t)e–xt dt t+ ∞ f (x) = ( ω(t)e–xt dt) ∞ ∞ s–t ∞ ∞ t–s –x(s+t) –x(t+s) dt) ds ds) dt ( t+ ω(t)ω(s)e ( s+ ω(s)ω(t)e ∞ ∞ = = –xt –xt ( ω(t)e dt) ( ω(t)e dt) =
(t
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
=
∞ ∞ ( ∞ ∞
=
Page 4 of 8
∞ ∞ t–s s–t ω(t)ω(s)e–x(s+t) dt) ds + ( s+ ω(s)ω(t)e–x(t+s) ds) dt t+ ∞ ( ω(t)e–xt dt) (s–t) ω(t)ω(s)e–x(t+s) dt ds (t+)(s+) ∞ ( ω(t)e–xt dt)
>
for all x > . Note that (.)-(.) and (.) lead to lim xK (x) = ,
x→
lim xK (x) = ,
x→
xK (x) – x = , x→ x→ (xK (x) – xK (x))
K (x) – x(K (x) – K (x)) lim f (x) = lim = lim x→∞ x→∞ x→∞ (K (x) – K (x))
(.)
lim f (x) = lim
+ o( x ) x + o( x ) x
Therefore, Lemma . follows easily from (.)-(.).
=
.
(.)
3 Main results Theorem . Let a, b ≥ . Then the double inequality √
< x+a
x e K (x) < √ π x+b
holds for all x > if and only if a ≥ / and b = . Proof Let x > , f (x) be defined by Lemma ., and f (x), f (x) and F(x) be respectively defined by f (x) =
π – xex K (x),
f (x) = ex K (x)
(.)
and F(x) =
π
– xex K (x) f (x) = . f (x) ex K (x)
(.)
Then from (.), (.) and (.) we clearly see that lim f (x) = lim f (x) = ,
(.)
f (x) –ex K (x) + xex K (x)[K (x) – K (x)] = = f (x). f (x) –ex K (x)[K (x) – K (x)]
(.)
x→∞
x→∞
It follows from (.)-(.), Lemmas . and . together with L’Hôpital’s rule that the function F(x) is strictly increasing on (, ∞) and lim F(x) =
x→∞
.
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 5 of 8
Note that (.) and (.) lead to lim F(x) = lim
x→
x→
π – x = . ex K (x)
(.)
Therefore, Theorem . follows easily from (.), (.), (.) and the monotonicity of F(x). Remark . From Lemma . we clearly see that the double inequality p
if p ≥ /, and inequality (.) is reversed if p = . Remark . also leads to Corollary .. Corollary . Let p, q ≥ . Then the double inequality
y + p y–x K (x) e < < x+p K (y)
y + q y–x e x+q
holds for all < x < y if and only if p ≥ / and q = .
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 6 of 8
Remark . We clearly see that the lower bound for K (x)/K (y) in Corollary . is better than the bounds given in (.) and (.) for ν = . Remark . From the inequality (x + ) < (x + ) x+
given in [], (.), and the fact that
x+
√ x > x +
√ for all x > we clearly see that the lower bound given in Theorem . for /πex K (x) is better than that given in (.) and (.). But the upper bound given in Theorem . is weaker than that given in (.). Remark . From Theorem . and Corollary . we clearly see that there exist θ = θ (x) ∈ (, /) and θ = θ (x) ∈ (, /) such that K (x) =
π e–x , (x + θ )
K (x) = +
(x + θ )
π e–x (x + θ )
for all x > . Theorem . Let x > , n ∈ N, Rn (x) = Kn+ (x)/Kn (x), u (x) = + /(x), v (x) = + /(x + /), and un (x) and vn (x) be defined by un (x) =
n + , vn– (x) x
vn (x) =
n + un– (x) x
(n ≥ ).
(.)
Then the double inequality vn (x) < Rn (x) =
Kn+ (x) < un (x) Kn (x)
(.)
holds for all x > and n ∈ N. Proof We use mathematical induction to prove inequality (.). From Corollary . we clearly see that inequality (.) holds for all x > and n = . Suppose that inequality (.) holds for n = k – (k ≥ ), that is, vk– (x) < Rk– (x) < uk– (x). Then it follows from (.) and (.) together with the formula Kν– (x) ν Kν+ (x) ν Kν (x) =– – =– + Kν (x) Kν (x) x Kν (x) x
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 7 of 8
given in [] that Rk (x) = vk (x) =
k + , Rk– (x) x uk– (x)
+
k k < Rk (x) < + = uk (x). x vk– (x) x
(.)
Inequality (.) implies that inequality (.) holds for n = k, and the proof of Theorem . is completed. Remark . Let n = , , . Then Theorem . leads to (x + ) K (x) x + x + < < , x(x + ) K (x) x(x + ) x + x + x + K (x) x + x + x + < < , x(x + x + ) K (x) x(x + ) (x + x + x + x + ) K (x) x + x + x + x + < < x(x + x + x + ) K (x) x(x + x + x + ) for all x > . Remark . It is not difficult to verify that
√
x
(x + ) + + > , x(x + ) x
x + x + < x(x + )
+
x + x
,
√ x + x + x + x + x + x + + x + > , < x(x + x + ) x x(x + ) √ (x + x + x + x + ) + x + > , x(x + x + x + ) x + x + x + x + x + x + < x(x + x + x + ) x
+
x +
x
,
for x > . Therefore, the bounds given in Remark . are better than the bounds given in (.) for ν = , , .
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China. 2 Customer Service Center, State Grid Zhejiang Electric Power Research Institute, Hangzhou, 310009, China. Acknowledgements The research was supported by the Natural Science Foundation of China under Grants 61673169, 61374086, 11371125 and 11401191. Received: 10 January 2017 Accepted: 8 February 2017
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 8 of 8
References 1. Abramowitz, M, Stegun, IA: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1970) 2. Luke, YL: Inequalities for generalized hypergeometric functions. J. Approx. Theory 5, 41-65 (1972) 3. Gaunt, RE: Inequalities for modified Bessel functions and their integrals. J. Math. Anal. Appl. 420(1), 373-386 (2014) 4. Segura, J: Bounds for ratios of modified Bessel functions and associated Turán-type inequalities. J. Math. Anal. Appl. 374(2), 516-528 (2011) 5. Bordelon, DJ, Ross, DK: Problem 72-75, ‘Inequalities for special functions’. SIAM Rev. 15, 665-670 (1973) 6. Paris, RB: An inequality for the Bessel function Jν (ν x). SIAM J. Math. Anal. 15(1), 203-205 (1984) 7. Laforgia, A: Bounds for modified Bessel functions. J. Comput. Appl. Math. 34(3), 263-267 (1991) 8. Baricz, Á: Bounds for modified Bessel functions of the first and second kinds. Proc. Edinb. Math. Soc. (2) 53(3), 575-599 (2010) 9. Anderson, GD, Vamanamurthy, MK, Vuorinen, M: Monotonicity rules in calculus. Am. Math. Mon. 113(9), 805-816 (2006) 10. Qi, F: Bounds for the ratio of two gamma functions. J. Inequal. Appl. 2010, Article ID 493085 (2010) 11. Watson, GN: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1995)
RESEARCH
Open Access
On approximating the modified Bessel function of the second kind Zhen-Hang Yang1,2 and Yu-Ming Chu1* *
Correspondence: [email protected] 1 School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article
Abstract In the article, we prove that the double inequalities √ –x √ –x πe πe < K0 (x) < √ , √ 2(x + a) 2(x + b)
1+
1 K1 (x) 1 < 0 if and only if a ≥ 1/4 and b = 0 if a, b ∈ [0, ∞), where Kν (x) is the modified Bessel function of the second kind. As applications, we provide bounds for Kn+1 (x)/Kn (x) with n ∈ N and present the necessary and sufficient condition such that √ the function x → x + pex K0 (x) is strictly increasing (decreasing) on (0, ∞). MSC: 33B10; 26A48 Keywords: modified Bessel function; gamma function; monotonicity
1 Introduction The modified Bessel function of the first kind Iν (x) is a particular solution of the secondorder differential equation x y (x) + xy (x) – x + ν y(x) = , and it can be expressed by the infinite series Iν (x) =
∞ n=
n+ν x . n!(ν + n + )
While the modified Bessel function of the second kind Kν (x) is defined by Kν (x) =
π(I–ν (x) – Iν (x)) , sin(πν)
(.)
where the right-hand side of the identity of (.) is the limiting value in case ν is an integer. The following integral representation formula and asymptotic formulas for the modified Bessel function of the second kind Kν (x) can be found in the literature [], .., .., .., ..:
∞
Kν (x) =
e–x cosh(t) cosh(νt) dt
(x > ),
(.)
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
K (x) ∼ – log x (x → ), –ν x (ν > , x → ), Kν (x) ∼ (ν)
ν – (ν – )(ν – ) π –x e + + + · · · Kν (x) ∼ x x !(x)
Page 2 of 8
(.) (.) (x → ∞).
(.)
From (.) we clearly see that
∞
K (x) =
–x cosh(t)
e
K (x) = –
∞
dt =
∞
e–xt dt, √ t –
te–xt dt = –K (x). √ t –
(.) (.)
Recently, the bounds for the modified Bessel function of the second kind Kν (x) have attracted the attention of many researchers. Luke [] proved that the double inequality √ x x + x < e K (x) < √ x + π (x + ) x
(.)
holds for all x > . Gaunt [] proved that the double inequality
x+
(x + ) < < (x + )
x e K (x) < √ π x
(.)
∞ takes place for all x > , where (x) = t x– e–t dx is the classical gamma function. In [], Segura proved that the double inequality ν+
√
x
x
+ ν
and ν ≥ . Bordelon and Ross [] and Paris [] provided the inequality ν Kν (x) y–x x >e Kν (y) y
(.)
for all ν > –/ and y > x > . Laforgia [] established the inequality –ν Kν (x) y–x x >e Kν (y) y for all y > x > if ν ∈ (, /), and inequality (.) is reversed if ν ∈ (/, ∞). Baricz [] presented the inequality –/ Kν (x) y–x x >e Kν (y) y for all y > x > and ν ∈ (–∞, –/) ∪ (/, ∞).
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 3 of 8
Motivated by inequality (.), in the article, we prove that the double inequality √ –x √ –x πe πe < K (x) < √ √ (x + a) (x + b) holds for all x > if and only if a ≥ / and b = if a, b ∈ [, ∞). As applications, we provide bounds for Kn+ (x)/Kn (x) with n ∈ N and present the necessary and sufficient condition √ such that the function x → x + pex K (x) is strictly increasing (decreasing) on (, ∞).
2 Lemmas In order to prove our main results, we need two lemmas which we present in this section. Lemma . (See []) Let –∞ ≤ a < b ≤ ∞, f , g : [a, b] → R be continuous on [a, b] and differentiable on (a, b), and g (x) = on (a, b). If f (x)/g (x) is increasing (decreasing) on (a, b), then so are the functions f (x) – f (b) . g(x) – g(b)
f (x) – f (a) , g(x) – g(a)
If f (x)/g (x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma . The function x → f (x) =
K (x) –x [K (x) – K (x)]
(.)
is strictly increasing from (, ∞) onto (, /). Proof Let ω(t) =
√
(t – )/(t + ). Then it follows from (.), (.) and (.) that
∞
K (x) – K (x) =
x K (x) – K (x) = –
ω(t)e–xt dt,
∞
ω(t)d e–xt
= ω(t)e–xt |t= t=∞
∞
+
ω (t)e–xt dt
∞
t– e–xt dt, – )/ ∞
ω(t) –xt K (x) – x K (x) – K (x) = e dt, t + ∞ ω(t) –xt e dt K (x) – x[K (x) – K (x)] = ∞ t+ –xt , f (x) = [K (x) – K (x)] ω(t)e dt ∞ tω(t) –xt ∞ ∞ ∞ – t+ e dt ω(t)e–xt dt + ω(t) e–xt dt tω(t)e–xt dt t+ ∞ f (x) = ( ω(t)e–xt dt) ∞ ∞ s–t ∞ ∞ t–s –x(s+t) –x(t+s) dt) ds ds) dt ( t+ ω(t)ω(s)e ( s+ ω(s)ω(t)e ∞ ∞ = = –xt –xt ( ω(t)e dt) ( ω(t)e dt) =
(t
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
=
∞ ∞ ( ∞ ∞
=
Page 4 of 8
∞ ∞ t–s s–t ω(t)ω(s)e–x(s+t) dt) ds + ( s+ ω(s)ω(t)e–x(t+s) ds) dt t+ ∞ ( ω(t)e–xt dt) (s–t) ω(t)ω(s)e–x(t+s) dt ds (t+)(s+) ∞ ( ω(t)e–xt dt)
>
for all x > . Note that (.)-(.) and (.) lead to lim xK (x) = ,
x→
lim xK (x) = ,
x→
xK (x) – x = , x→ x→ (xK (x) – xK (x))
K (x) – x(K (x) – K (x)) lim f (x) = lim = lim x→∞ x→∞ x→∞ (K (x) – K (x))
(.)
lim f (x) = lim
+ o( x ) x + o( x ) x
Therefore, Lemma . follows easily from (.)-(.).
=
.
(.)
3 Main results Theorem . Let a, b ≥ . Then the double inequality √
< x+a
x e K (x) < √ π x+b
holds for all x > if and only if a ≥ / and b = . Proof Let x > , f (x) be defined by Lemma ., and f (x), f (x) and F(x) be respectively defined by f (x) =
π – xex K (x),
f (x) = ex K (x)
(.)
and F(x) =
π
– xex K (x) f (x) = . f (x) ex K (x)
(.)
Then from (.), (.) and (.) we clearly see that lim f (x) = lim f (x) = ,
(.)
f (x) –ex K (x) + xex K (x)[K (x) – K (x)] = = f (x). f (x) –ex K (x)[K (x) – K (x)]
(.)
x→∞
x→∞
It follows from (.)-(.), Lemmas . and . together with L’Hôpital’s rule that the function F(x) is strictly increasing on (, ∞) and lim F(x) =
x→∞
.
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 5 of 8
Note that (.) and (.) lead to lim F(x) = lim
x→
x→
π – x = . ex K (x)
(.)
Therefore, Theorem . follows easily from (.), (.), (.) and the monotonicity of F(x). Remark . From Lemma . we clearly see that the double inequality p
if p ≥ /, and inequality (.) is reversed if p = . Remark . also leads to Corollary .. Corollary . Let p, q ≥ . Then the double inequality
y + p y–x K (x) e < < x+p K (y)
y + q y–x e x+q
holds for all < x < y if and only if p ≥ / and q = .
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 6 of 8
Remark . We clearly see that the lower bound for K (x)/K (y) in Corollary . is better than the bounds given in (.) and (.) for ν = . Remark . From the inequality (x + ) < (x + ) x+
given in [], (.), and the fact that
x+
√ x > x +
√ for all x > we clearly see that the lower bound given in Theorem . for /πex K (x) is better than that given in (.) and (.). But the upper bound given in Theorem . is weaker than that given in (.). Remark . From Theorem . and Corollary . we clearly see that there exist θ = θ (x) ∈ (, /) and θ = θ (x) ∈ (, /) such that K (x) =
π e–x , (x + θ )
K (x) = +
(x + θ )
π e–x (x + θ )
for all x > . Theorem . Let x > , n ∈ N, Rn (x) = Kn+ (x)/Kn (x), u (x) = + /(x), v (x) = + /(x + /), and un (x) and vn (x) be defined by un (x) =
n + , vn– (x) x
vn (x) =
n + un– (x) x
(n ≥ ).
(.)
Then the double inequality vn (x) < Rn (x) =
Kn+ (x) < un (x) Kn (x)
(.)
holds for all x > and n ∈ N. Proof We use mathematical induction to prove inequality (.). From Corollary . we clearly see that inequality (.) holds for all x > and n = . Suppose that inequality (.) holds for n = k – (k ≥ ), that is, vk– (x) < Rk– (x) < uk– (x). Then it follows from (.) and (.) together with the formula Kν– (x) ν Kν+ (x) ν Kν (x) =– – =– + Kν (x) Kν (x) x Kν (x) x
(.)
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
Page 7 of 8
given in [] that Rk (x) = vk (x) =
k + , Rk– (x) x uk– (x)
+
k k < Rk (x) < + = uk (x). x vk– (x) x
(.)
Inequality (.) implies that inequality (.) holds for n = k, and the proof of Theorem . is completed. Remark . Let n = , , . Then Theorem . leads to (x + ) K (x) x + x + < < , x(x + ) K (x) x(x + ) x + x + x + K (x) x + x + x + < < , x(x + x + ) K (x) x(x + ) (x + x + x + x + ) K (x) x + x + x + x + < < x(x + x + x + ) K (x) x(x + x + x + ) for all x > . Remark . It is not difficult to verify that
√
x
(x + ) + + > , x(x + ) x
x + x + < x(x + )
+
x + x
,
√ x + x + x + x + x + x + + x + > , < x(x + x + ) x x(x + ) √ (x + x + x + x + ) + x + > , x(x + x + x + ) x + x + x + x + x + x + < x(x + x + x + ) x
+
x +
x
,
for x > . Therefore, the bounds given in Remark . are better than the bounds given in (.) for ν = , , .
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 School of Mathematics and Computation Sciences, Hunan City University, Yiyang, 413000, China. 2 Customer Service Center, State Grid Zhejiang Electric Power Research Institute, Hangzhou, 310009, China. Acknowledgements The research was supported by the Natural Science Foundation of China under Grants 61673169, 61374086, 11371125 and 11401191. Received: 10 January 2017 Accepted: 8 February 2017
Yang and Chu Journal of Inequalities and Applications (2017) 2017:41
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