Figure 1.
Plot illustration for the expression (2.6).
Citation: Damian G. D'Souza. The Parathyroid Hormone Family of Ligands and Receptors[J]. AIMS Medical Science, 2015, 2(3): 118-130. doi: 10.3934/medsci.2015.3.118
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Many research papers have studied the properties of convex functions that make this concept interesting in mathematical analysis [1,2,3,4]. In recent years, important generalizations have been made in the context of convexity: quasi-convex [5], pseudo-convex [6], invex and preinvex [7], strongly convex [8], approximately convex [9], $ MT $-convex [10], $ (\alpha, m) $-convex [11], and strongly $ (s, m) $-convex [12,13,14,15]. Here, we recall the notion of convexity: A function $ g:[\beta_1, \beta_2]\subset \mathbb{R}\to \mathbb{R} $ is said to be convex if the following inequality holds
|
$ g(tx+(1−t)y)≤tg(x)+(1−t)g(y),x,y∈[β1,β2],t∈[0,1]. $
|
(1.1) |
Now, we recall our basic definition, the so-called quasi-convex function.
Definition 1.1 ([16]). A function $ g:[\beta_1, \beta_2]\to \mathbb{R} $ is said quasi-convex on $ [\beta_1, \beta_2] $ if
|
$ g(tx+(1−t)y)≤max{g(x),g(y)}, $
|
(1.2) |
for any $ x, y\in [\beta_1, \beta_2] $ and $ t \in [0, 1] $.
It is important to note that, any convex function is a quasi-convex but the reverse is not true. In the following example we explain that fact.
Example 1.1 ([5]). The function $ h:[-2, 2]\to \mathbb{R} $, defined by
|
$ h(s)={1, for s∈[−2,−1],s2, for s∈(−1,2], $
|
is not convex on $ [-2, 2] $ but it is easy to see that the function is quasi-convex on $ [-2, 2] $.
Notice that $ h $ is quasi-convex if and only if all the level sets of $ h $ are intervals (convex sets of the line).
The use of the convex function to study the integral inequalities have been deeply investigated, especially for the well-known inequality of Hermite-Hadamard type (HH-type inequality). The HH-type inequalities are one of the most important type inequalities and have a strong relationship to convex functions. In 1893 Hermite and Hadamard [17] found independently that for any convex function $ g:[\beta_1, \beta_2]\to \mathbb{R} $, the inequality
|
$ g(β1+β22)≤1β2−β1∫β2β1g(x)dx≤g(β1)+g(β2)2, $
|
(1.3) |
holds.
In the field of mathematical analysis, many scholars have focused on defining new convexity and implementing of the problems based on their features. The features that make the results different from each other include lower and higher order derivative of the function. The differential equations with impulse perturbations lie in a special significant position in the theory of differential equations. Among them, integral inequality methods are the important tools to investigate the qualitative characteristics of solutions of different kinds of equations such as differential equations, difference equations, partial differential equations, and impulsive differential equations; see [18,19,20,21,22,23] for more details.
The HH-type inequality (1.3) has been applied to various convex functions like s-geometrically convex functions [24], GA-convex functions [25], $ MT $-convex function [10], $ (\alpha, m) $-convex functions [26] and many other types can be found in [27]. Besides, the HH-type inequality (1.3) has been applied to a numerous type of convex functions in the sense of fractional calculus like $ F $-convex functions [28], $ \lambda_{\psi} $-convex functions [29], $ MT $-convex functions [30], $ (\alpha, m) $-convex functions [11], new class of convex functions [31] and many other types can be found in the literature. Meanwhile, it has been applied to other models of fractional calculus like standard Riemann-Liouville fractional operators [32,33], conformable fractional operators [34,35,36], generalized fractional operators [37], $ \psi $-RL-fractional operators [38,39], Tempered fractional operators [40], and AB- and Prabhakar fractional operators [41].
In view of the above indices, we extend the work done in [42] to establish some modified HH-type inequalities for the 3-times differentiable quasi-convex functions.
This section deals with our main results. Throughout this paper, we mean $ g\in L[\beta_1, \beta_2] $ that the function $ g $ is differential and continuous on $ [\beta_1, \beta_2] $.
Lemma 2.1. Suppose that $ g:J\subset \mathbb{R}\to \mathbb{R} $ is a differentiable function such that $ \beta_1, \beta_2 \in J $ with $ \beta_1 < \beta_2 $. If $ g'''\in L[\beta_1, \beta_2] $, then we have
|
$ g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)=(β2−β1)396[∫10t3g‴(tβ1+β22+(1−t)β1)dt+∫10(t−1)3g‴(tβ2+(1−t)β1+β22)dt]. $
|
(2.1) |
Proof. By applying integration by parts three times to get
|
$ J1:=∫10t3g‴(tβ1+β22+(1−t)β1)dt=2β2−β1g″(β1+β22)−6β2−β1∫10t2g″(tβ1+β22+(1−t)β1)dt=2β2−β1g″(β1+β22)−12(β2−β1)2g′(β1+β22)+24(β2−β1)2∫10tg′(tβ1+β22+(1−t)β1)dt=2β2−β1g″(β1+β22)−12(β2−β1)2g′(β1+β22)+48(β2−β1)3g(β1+β22)−48(β2−β1)3∫10g(tβ1+β22+(1−t)β1)dt. $
|
(2.2) |
Making use of change of the variable $ x = t\frac{\beta_1+\beta_2}{2}+(1-t)\beta_1 $ for $ t\in [0, 1] $ and multiplying by $ \frac{(\beta_2-\beta_1)^3}{96} $ on both sides, we obtain
|
$ (β2−β1)396J1=(β2−β1)248g″(β1+β22)−β2−β18g′(β1+β22)+12g(β1+β22)−1β2−β1∫β1+β22β1g(x)dx. $
|
(2.3) |
Analogously, we can deduce
|
$ (β2−β1)396J2:=(β2−β1)396∫10(t−1)3g‴(tβ2+(1−t)β1+β22)dt=(β2−β1)248g″(β1+β22)+β2−β18g′(β1+β22)+12g(β1+β22)−1β2−β1∫β2β1+β22g(x)dx. $
|
(2.4) |
Finally, by adding (2.3) and (2.4), we get the required identity (2.1).
Remark 2.1. Notice that f being quasi-convex is not equivalent to $ |f| $ being quasi-convex. For instance, $ g(x) = x^2-1 $ is only quasi-convex (but not $ |g(x)| $), whereas $ g(x) = 1 $ if $ x\in[-1, 1] $ and $ g(x) = -1 $ otherwise, is not quasi-convex, but $ |g(x)| = 1 $ is quasi-convex.
Theorem 2.1. Suppose that $ g:J\subseteq [0, +\infty)\to \mathbb{R} $ is a differentiable function such that $ g'''\in L[\beta_1, \beta_2] $, where $ \beta_1, \beta_2\in J $ with $ \beta_1 < \beta_2 $. If $ |g'''| $ is quasi-convex function on $ [\beta_1, \beta_2] $, then we have
|
$ |g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|≤(β2−β1)3384K, $
|
(2.5) |
where $ K = \max\left\{\left|g'''\left(\frac{\beta_1+\beta_2}{2}\right)\right|, \left|g'''(\beta_1)\right|\right\}+ \max\left\{\left|g'''\left(\frac{\beta_1+\beta_2}{2}\right)\right|, \left|g'''(\beta_2)\right|\right\}. $
Proof. Making use of Lemma 2.1 and the quasi-convexity $ |g'''| $, we have that
|
$ |g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|≤(β2−β1)396[∫10t3|g‴(tβ1+β22+(1−t)β1)|dt+∫10(t−1)3|g‴(tβ2+(1−t)β1+β22)|dt]≤(β2−β1)396∫10t3max{|g‴(β1+β22)|,|g‴(β1)|}dt+(β2−β1)396∫10(1−t)3max{|g‴(β1+β22)|,|g‴(β2)|}dt≤(β2−β1)3384[max{|g‴(β1+β22)|,|g‴(β1)|}+max{|g‴(β1+β22)|,|g‴(β2)|}]. $
|
This rearranges to the desired result.
Example 2.1. To clarify the following expression occurs in Theorem 2.1
|
$ p:=(β2−β1)224g″(β1+β22), $
|
(2.6) |
we consider the function $ g(x) = \frac{x}{x^2+2} $ on the interval $ [\beta_1, \beta_2] = [0, 1] $. Then, we have
|
$ y(x):=g″(x)=2x(4x2−3)(x2+2)2;p=124g″(12)=−462187. $
|
Figure 1 demonstrates the intersections and relationships between the functions $ g(x), y(x) $ and the point $ p $ geometrically.
Corollary 2.1. Let the assumptions of Theorem 2.1 be valid and let
|
$ H:=|g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|. $
|
Then,
(i) if $ |g'''| $ is increasing, then we have
|
$ H≤(β2−β1)3384[|g‴(β2)|+|g‴(β1+β22)|], $
|
(2.7) |
(ii) if $ |g'''| $ is decreasing, then we have
|
$ H≤(β2−β1)3384[|g‴(β1)|+|g‴(β1+β22)|], $
|
(2.8) |
(iii) if $ g'''\left(\frac{\beta_1+\beta_2}{2}\right) = 0 $, then we have
|
$ H≤(β2−β1)3384[|g‴(β1)|+|g‴(β2)|], $
|
(2.9) |
(iv) if $ g'''(\beta_1) = g'''(\beta_2) = 0 $, then we have
|
$ H≤(β2−β1)3384|g‴(β1+β22)|. $
|
(2.10) |
Theorem 2.2. Suppose that $ g:J\subseteq [0, +\infty)\to \mathbb{R} $ is a differentiable function such that $ g'''\in L[\beta_1, \beta_2] $, where $ \beta_1, \beta_2\in J $ with $ \beta_1 < \beta_2 $. If $ |g'''|^q $ is quasi-convex function on $ [\beta_1, \beta_2] $ and $ q > 1 $ with $ \frac{1}{p}+\frac{1}{q} = 1 $, then we have
|
$ |g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|≤(β2−β1)396(3p+1)1pKq, $
|
(2.11) |
where $ K_q = \left(\max\left\{\left|g'''\left(\frac{\beta_1+\beta_2}{2}\right)\right|^q, |g'''(\beta_1)|^q\right\}\right)^{\frac{1}{q}}+\left(\max\left\{\left|g'''\left(\frac{\beta_1+\beta_2}{2}\right)\right|^q, |g'''(\beta_2)|^q\right\}\right)^{\frac{1}{q}} $.
Proof. Let $ p > 1 $. Then from Lemma 2.1 and using the Hölder inequality, we can deduce
|
$ |g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|≤(β2−β1)396[∫10t3|g‴(tβ1+β22+(1−t)β1)|dt+∫10(1−t)3|g‴(tβ2+(1−t)β1+β22)|dt]≤(β2−β1)396(∫10t3pdt)1p(∫10|g‴(tβ1+β22+(1−t)β1)|qdt)1q+(β2−β1)396(∫10(1−t)3pdt)1p(∫10|g‴(tb+(1−t)β1+β22)|qdt)1q. $
|
The quasi-convexity of $ |g'''|^q $ on $ [\beta_1, \beta_2] $ implies that
|
$ ∫10|g‴(tβ1+β22+(1−t)β1)|qdt≤max{|g‴(β1+β22)|q,|g‴(β1)|q}, $
|
and
|
$ ∫10|g‴(tb+(1−t)β1+β22)|qdt≤max{|g‴(β1+β22)|q,|g‴(β2)|q}. $
|
Therefore, we obtain
|
$ |g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|≤(β2−β1)396(3p+1)1pKq, $
|
where we used the identities
|
$ ∫10t3pdt=∫10(1−t)3pdt=13p+1. $
|
Thus, our proof is completely done.
Corollary 2.2. Let the assumptions of Theorem 2.2 be valid and let
|
$ H=|g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|. $
|
Then,
(i) if $ |g'''| $ is increasing, then we have
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$ H≤(β2−β1)396(3p+1)1p[|g‴(β2)|+|g‴(β1+β22)|], $
|
(2.12) |
(ii) if $ |g'''| $ is decreasing, then we have
|
$ H≤(β2−β1)396(3p+1)1p[|g‴(β1)|+|g‴(β1+β22)|], $
|
(2.13) |
(iii) if $ g'''\left(\frac{\beta_1+\beta_2}{2}\right) = 0 $, then we have
|
$ H≤(β2−β1)396(3p+1)1p[|g‴(β1)|+|g‴(β2)|], $
|
(2.14) |
(iv) if $ g'''(\beta_1) = g'''(\beta_2) = 0 $, then we have
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$ H≤(β2−β1)396(3p+1)1p|g‴(β1+β22)|. $
|
(2.15) |
Theorem 2.3. Suppose that $ g:J\subseteq [0, +\infty)\to \mathbb{R} $ is a differentiable function such that $ g'''\in L[\beta_1, \beta_2], $ where $ \beta_1, \beta_2\in J $ with $ \beta_1 < \beta_2 $. If $ |g'''|^q $ is quasi-convex function on $ [\beta_1, \beta_2] $ and $ q\geq 1 $ with $ \frac{1}{p}+\frac{1}{q} = 1 $, then we have
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$ |g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|≤(β2−β1)3384Kq, $
|
(2.16) |
where $ K_q $ is as before.
Proof. From Lemma 2.1, properties of modulus, and power mean inequality, we have
|
$ |g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|≤(β2−β1)396[∫10t3|g‴(tβ1+β22+(1−t)β1)|dt+∫10(1−t)3|g‴(tβ2+(1−t)β1+β22)|dt]≤(β2−β1)396(∫10t3dt)1p(∫10t3|g‴(tβ1+β22+(1−t)β1)|qdt)1q+(β2−β1)396(∫10(1−t)3dt)1p(∫10(1−t)3|g‴(tb+(1−t)β1+β22)|qdt)1q. $
|
Then, by using the quasi-convexity of $ |g'''|^q $ on $ [\beta_1, \beta_2] $, we have
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$ ∫10t3|g‴(tβ1+β22+(1−t)β1)|qdt≤14max{|g‴(β1+β22)|q,|g‴(β1)|q}, $
|
and
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$ ∫10(1−t)3|g‴(tb+(1−t)β1+β22)|qdt≤14max{|g‴(β1+β22)|q,|g‴(β2)|q}. $
|
Therefore, we obtain
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$ |g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|≤(β2−β1)3384Kq, $
|
Hence, our proof is completely done.
Remark 2.2. If $ q = \frac{p}{p-1} \ (p > 1) $, the constants of Theorem 2.2 are improved, since $ \frac{1}{(3p+1)^{\frac{1}{p}}} < 1 $.
The following corollary improves the inequalities (2.7)–(2.10).
Corollary 2.3. Let the assumptions of Theorem 2.3 be valid and let
|
$ H=|g(β1+β22)−1β2−β1∫β2β1g(x)dx+(β2−β1)224g″(β1+β22)|. $
|
Then,
(i) If $ |g'''| $ is increasing, we obtain (2.7).
(ii) If $ |g'''| $ is decreasing, we obtain (2.8).
(iii) If $ g'''\left(\frac{\beta_1+\beta_2}{2}\right) = 0 $, we obtain (2.9),
(iv) if $ g'''(\beta_1) = g'''(\beta_2) = 0 $, we obtain (2.10).
Consider the special means of positive real numbers $ \beta_1 > 0 $ and $ \beta_2 > 0 $, define by:
● Arithmetic Mean:
|
$ A(β1,β2)=β1+β22. $
|
● Logarithmic mean:
|
$ L(β1,β2)=β2−β1ln|β2|−ln|β1|, |β1|≠|β2|, β1,β2≠0. $
|
● Generalized log-mean:
|
$ Lp(β1,β2)=[β2p+1−β1p+1(p+1)(β2−β1)]1p, p∈Z∖{−1,0}, β1≠β2. $
|
Remark 3.1. Let $ 0 < \alpha\le 1 $ and $ x > 0 $. Then, we consider
|
$ g(x)=xα+3(α+1)(α+2)(α+3),g‴(x)=xα. $
|
For each $ x, y > 0 $ and $ t\in[0, 1] $, we see that $ (tx+(1-t)y)^{\alpha}\le t^{\alpha}x^{\alpha}+(1-t)^{\alpha}y^{\alpha} $, then we see that $ g'''(x) $ is $ \alpha $-convex function on $ (0, +\infty) $ and $ g\left(\frac{\beta_1+\beta_2}{2}\right) = \frac{A^{\alpha+3}(\beta_1, \beta_2)}{(\alpha +1)(\alpha+2)(\alpha+3)} $. Furthermore, we have
|
$ 1β2−β1∫β2β1g(x)dx=1(α+1)(α+2)(α+3)[βα+42−βα+41(α+4)(β2−β1)]=1(α+1)(α+2)(α+3)Lα+3α+3(β1,β2). $
|
Above we used the definition of $ \alpha $-convexity [11]: A function $ g:[0, r]\to\mathbb{R}, r > 0 $ is said to be $ \alpha $-convex, if the following holds:
|
$ g(tx+(1−t)y)≤tαg(x)+(1−tα)g(y),x,y∈[0,r],t,α∈[0,1]. $
|
Proposition 3.1. Let $ 0 < \alpha\leq 1 $ and$ \beta_1, \beta_2\in \mathbb{R}^+ $ with $ \beta_1 < \beta_2 $, then we have
|
$ 384(α+1)(α+2)(α+3)|Aα+3(β1,β2)−Lα+3α+3(β1,β2)+(β2−β1)2(α+2)(α+3)24Aα+1(β1,β2)|≤(β2−β1)3[max{g(β1+β22)α,βα1}+max{g(β1+β22)α,βα2}]. $
|
Proof. Since $ x^{\alpha} $ is quasi-convex for each $ x > 0 $ and $ \alpha\in(0, 1) $ because every non-decreasing continuous function is also quasi-convex, so the assertion follows from inequality (2.5) with $ g(x) = \frac{x^{\alpha+3}}{(\alpha+1)(\alpha+2)(\alpha+3)} $.
Proposition 3.2. Let $ \beta_1, \beta_2\in \mathbb{R} $ such that $ \beta_1 < \beta_2 $ and $ [\beta_1, \beta_2]\subset (0, +\infty) $, then we have
|
$ |A−1(β1,β2)−L−1(β1,β2)+(β2−β1)212A−3(β1,β2)|≤(β2−β1)364[β−41+g(β1+β22)−4]. $
|
Proof. The assertion follows from inequality (2.11) with $ g(x) = \frac{1}{x}, \ x\in [\beta_1, \beta_2] $.
Proposition 3.3. Let $ \beta_1, \beta_2\in \mathbb{R} $ with $ 0 < \beta_1 < \beta_2 $ and $ n\in \mathbb{N} $, $ k\ge 5 $, then for all $ q\ge 1 $, we have
|
$ |Ak(β1,β2)−Lkk(β1,β2)+(β2−β1)2k!24(k−2)!Ak−2(β1,β2)|≤k!(β2−β1)32k+23(3p+1)1p(k−3)![(β1+β2)k−3+2k−3βk−32]. $
|
Proof. Let $ g(x) = x^k, \ x\in [\beta_1, \beta_2], \ k\in \mathbb{N} $ with $ k\ge 5 $, then we have
|
$ g‴(x)=k!(k−3)!xk−3, $
|
and it is easy to see that $ g''' $ is an increasing and quasi-convex function. Then, by applying Corollary 2.2(ⅰ) above, we have
|
$ |Ak(β1,β2)−Lkk(β1,β2)+(β2−β1)2k!24(k−2)!Ak−2(β1,β2)|≤(β2−β1)396(3p+1)1pK, $
|
where
|
$ K=k!(k−3)![max{|β1+β22|k−3,|β1|k−3}+max{|β1+β22|k−3,|β2|k−3}]. $
|
Then, by applying Corollary 2.1(9) to $ g $ above, we get
|
$ |Ak(β1,β2)−Lkk(β1,β2)+(β2−β1)2k!24(k−2)!Ak−2(β1,β2)|≤k!(β2−β1)32k+23(3p+1)1p(k−3)![(β1+β2)k−3+2k−3βk−32], $
|
and this completes the proof.
Here, we consider two particular functions.
● First, we define $ g:\mathbb{R}\to \mathbb{R} $, by $ g(x) = e^x $.
Then, we have $ g'''(x) = e^x $ and
|
$ ||g‴||∞=supt∈[β1,β2]|g‴(t)|=eβ2, $
|
By applying inequality (2.7) for above $ ||g'''||_{\infty} $, we can deduce
|
$ |(1+(β2−β1)224)eβ1+β22−1β2−β1(eβ2−eβ1)|≤(β2−β1)3384||g‴||∞=(β2−β1)3384eβ2. $
|
Particularly for $ \beta_1 = 0 $, it follows that
|
$ |(1+β2224)eβ22−1β2(eβ2−1)|≤β23384eβ2, $
|
(3.1) |
and for $ \beta_2 = 1 $, it follows that
|
$ |2524√e−e+1|≤e384. $
|
● Now, we define $ g:\mathbb{R^+}\to \mathbb{R} $, by $ g(x) = \frac{1}{x} $.
Then, since $ |g'''(x)| = \frac{6}{x^4} $ is quasi-convex in $ [\beta_1, \beta_2]\subset \mathbb{R}^+ $ and
|
$ ||g‴||∞=supt∈[β1,y]|g‴(t)|=6β14, 0<β1<y. $
|
By applying inequality (2.8) to the function $ g $ above, we get
|
$ |2β1+β2−lnβ2−lnβ1β2−β1+(β2−β1)212(2β1+β2)3|≤(β2−β1)33842||g‴||∞=(β2−β1)332β14. $
|
(3.2) |
In view of (3.2) and Proposition 2, we can deduce
|
$ |A−1(β1,β2)−L−1(β1,β2)+(β2−β1)212A−3(β1,β2)|≤(β2−β1)364[β−41+g(β1+β22)−4]≤(β2−β1)332β14. $
|
For further illustration on the inequalities (3.1) and (3.2), we present some plot examples. Figures 2 and 3 illustrate the inequalities (3.1) and (3.2), respectively.
Let
|
$ k(β2)=(1+β2224)eβ22−1β2(eβ2−1),K(β2)=β32384eβ2, $
|
and
|
$ h(β1,β2)=2β1+β2−lnβ2−lnβ1β2−β1+(β2−β1)212(2β1+β2)3,H(β1,β2)=(β2−β1)332β41. $
|
Furthermore, Figures 4 and 5 show $ K(\beta_2)-k(\beta_2) $ and $ D(\beta_1, \beta_2):= H(\beta_1, \beta_2)-h(\beta_1, \beta_2) $, receptively. From Figure 5, we can see that all values of $ D(\beta_1, \beta_2) $ are positive which confirms the validity of (3.2).
In this paper we have established new Hermite–Hadamard inequality mainly motivated by Alomari et al in [42] for quasi-convex functions with $ g\in C^3([\beta_1, \beta_2]) $ such that $ g'''\in L([\beta_1, \beta_2]) $ and we give some applications to some special means and for some particular functions. We hope that the ideas used in this paper may inspire interested readers to explore some new applications.
We believe that our results, this new understanding of Hermite-Hadamard integral inequalities for quasi-convex functions, will be vital information for the future studies of these models of integral inequality. One can obtain the similar results for other kind of convex functions
We would like to express our special thanks to the editor and referees. Also, the third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
The authors declare no conflict of interest.
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Damian G. D'Souza. The Parathyroid Hormone Family of Ligands and Receptors[J]. AIMS Medical Science, 2015, 2(3): 118-130. doi: 10.3934/medsci.2015.3.118
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