# Coordinated MT-s1,s2-Convex Functions and Their Integral Inequalities of Hermite–Hadamard Type

Authors: Hua Mei; Aying Wan; Bai-Ni Guo
Journal: Journal of Mathematics
 (2021)

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## Abstract

In the paper, the authors introduce a new concept of MT-s1,s2-convex functions on the coordinates on the rectangle of the plane and establish some new Hermite–Hadamard-type inequalities for this kind of functions.


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## Body

## 1. Motivations

At first, we recall several kinds of convex functions as follows.Definition 1.
(see [1]). Let I⊆ℝ be an interval. A nonnegative function f:I⟶ℝ0=0,∞ is said to be MT-convex if the inequality(1)ftx+1−ty≤t21−tfx+1−t2tfyholds for all x,y∈I and t∈0,1.Definition 2.
(see [2, 3]). Let s∈0,1 be a real number. A function f:ℝ⟶ℝ0 is said to be s-convex in the second sense if(2)ftx+1−ty≤tsfx+1−tsfy,for all x,y∈I and t∈0,1.Definition 3.
(see [4, 5]). A function f:Δ=a,b×c,d⊆ℝ2⟶ℝ is said to be convex on the coordinates on Δ if(3)ftx+1−tz,λy+1−λw≤tλfx,y+t1−λfx,w+1−tλfz,y+1−t1−λfz,wholds for all t,λ∈0,1 and x,y,z,w∈Δ. If the inequality (3) is reversed, then f is said to be concave on the coordinates on Δ.Definition 4.
(see [6]). We say that a function f:Δ=a,b×c,d⊆ℝ2⟶ℝ0 is MT-convex on the coordinates on Δ with a<b and c<d, if the inequality(4)ftx+1−tz,λy+1−λw≤tλ41−t1−λfx,y+t1−λ41−tλfx,w+1−tλ4t1−λfz,y+1−t1−λ4tλfz,wis valid for all t,λ∈0,1 and x,y,z,w∈Δ.
In the papers [7–12] and closely related references therein, the HT-convexity, GT-convexity, and the s;m-P-convexity on the coordinates were introduced and investigated.
Combining the structures of Definitions2 and 4, we introduce the notion of coordinated MT-s1,s2-convex functions as follows.Definition 5.
Fors1,s2∈0,12, a function f:Δ=a,b×c,d⊆ℝ2⟶ℝ0 is said to be MT-s1,s2-convex on the coordinates on Δ with a<b and c<d, if the inequality(5)ftx+1−tz,λy+1−λw≤ts1/2λs2/241−t1−λfx,y+ts1/21−λs2/241−tλfx,w+1−ts1/2λs2/24t1−λfz,y+1−ts1/21−λs2/24tλfz,wholds for all t,λ∈0,1 and x,y,z,w∈Δ. If inequality (5) is reversed, then f is said to be a MT-s1,s2-concave function on the coordinates on Δ.


## 2. Simple Properties of MT-s1,s2-Convex Functions

After introduced Definition5, now we are in a position to investigate in this section simple properties of MT-s1,s2-convex functions on the coordinates on Δ.Proposition 1.
Lets1,s2∈0,12 and f:Δ=a,b×c,d⊆ℝ2⟶ℝ0. If f is nonnegative and convex on the coordinates on Δ, then f is MT-convex on the coordinates on Δ, while f is also MT-s1,s2-convex on the coordinates on Δ.Proof.
This follows fromts/2//21−t≥t1/2/21−t≥t for s∈0,1 and t∈0,1.Example 1.
(see [13], p. 104). When p∈0,1/1000, the functions fx=xp and gx=1+xp for x∈Δ1=1,∞ are MT-convex, but they are not convex on Δ1.
Form∈0,1/100, the function hx=1+x2m for x∈1,3/2 is MT-convex, but it is not convex on 1,3/2.Example 2.
(see [6], p. 259). When p∈0,1/1000, the function fx,y=xp+yp for x,y∈Δ1 is MT-convex on the coordinates on Δ1, but it is not convex on the coordinates on Δ1.Remark 1.
We now discuss Examples1 and 2 mentioned above.
For0<p<1, c≥0, and I⊆0,∞ being a nonempty interval, the function φx=x+cp is concave on I. Therefore, for t=1/2 and all x,y∈I with x≠y, we have(6)φtx+1−ty=x+y2+cp>x+cp+y+cp2=t21−tφx+1−t2tφy.
Accordingly, the functionφx is not MT-convex on I. From this, we conclude as follows:(1)
Forp∈0,1/1000, the functions fx=xp and gx=1+xp with respect to x∈Δ1 are not MT-convex on Δ1(2)
Form∈0,1/100, the function hx=1+x2m with respect to x∈1,3/2 is not MT-convex, but it is convex on 1,1/1−2m and is concave on 1/1−2m,3/2(3)
Forp∈0,1/1000, the function fx,y=xp+yp with respect to x,y∈Δ1 is not MT-convex on the coordinates on Δ1Proposition 2.
Letfx,y=x1/2 for x,y∈ℝ+2=0,∞2. Then, the function fx,y is MT-0.5,0.02-convex, but not MT-convex, on the coordinates on ℝ+2.Proof.
Fors1=0.5 and s2=0.02, for t,λ∈0,1, and for x,y,z,w∈ℝ+2, by Definition 5, we deduce(7)ftx+1−tz,λy+1−λw=tx+1−tz1/2t0.5/2λ0.02/241−t1−λfx,y+t0.5/21−λ0.02/241−tλfx,w+1−t0.5/2λ0.02/24t1−λfz,y+1−t0.5/21−λ0.02/24tλfz,w=λ0.02/221−λ+1−λ0.02/22λt0.5/221−tx1/2+1−t0.5/22tz1/2.
Making use of the inequalityλ0.02/2/21−λ+1−λ0.02/2/2λ≥20.49>1.4 and letting u=x/z result in(8)tx+1−tz1/2=z1/2tu+1−t1/2<1.4z1/2t0.5/221−tu1/2+1−t0.5/22t<λ0.02/221−λ+1−λ0.02/22λt0.5/221−tx1/2+1−t0.5/22tz1/2.
This means that the functionfx,y is MT-0.5,0.02-convex on the coordinates on ℝ+2.
Forx,y,z,w∈ℝ+2 with x≠z, taking t=λ=1/2 in Definition 5 leads to(9)ftx+1−tz,λy+1−λw=x+y21/2>x1/2+z1/22=tλ41−t1−λfx,y+t1−λ41−tλfx,w+1−tλ4t1−λfz,y+1−t1−λ4tλfz,w.
This means that the functionfx,y is not MT-convex on the coordinates on ℝ+2. The proof of Proposition 2 is complete.


## 3. A Lemma

In order to establish integral inequalities of the Hermite–Hadamard type for MT-s1,s2-convex functions on the coordinates on Δ, we need the following lemma.Lemma 1.
Letf:Δ=a,b×c,d⊆ℝ2⟶ℝ have partial derivatives of the second order and let a<b and c<d. If ∂2f/∂x∂y∈L1Δ, then(10)If=fb,d−1b−a∫abfx,ddx−1d−c∫cdfb,ydy+1b−ad−c∫cd∫abfx,ydxdy=b−ad−c∫01∫011−t1−λ∂2fta+1−tb,λc+1−λd∂x∂ydtdλ.Proof.
Integrating by parts gives(11)∫01∫011−t1−λ∂2fta+1−tb,λc+1−λd∂x∂ydtdλ=1a−b∫011−λ1−t∂fta+1−tb,λc+1−λd∂y|t=0t=1+∫01∂fta+1−tb,λc+1−λd∂ydtdλ=1a−b∫01λ−1∂fb,λc+1−λd∂ydλ+∫01∫011−λ∂fta+1−tb,λc+1−λd∂ydtdλ=1a−bc−dλ−1fb,λc+1−λd|λ=1λ=0−∫01fb,λc+1−λddλ+∫011−λfta+1−tb,λc+1−λd|λ=1λ=0dt+∫01∫01fta+1−tb,λc+1−λddtdλ=1a−bc−dfb,d−∫01fb,λc+1−λddλ−∫01fta+1−tb,ddt+∫01∫01fta+1−tb,λc+1−λddtdλ=fb,d−1b−a∫abfx,ddx−1d−c∫cdfb,ydy+1b−ad−c∫cd∫abfx,ydxdy.
The proof of Lemma1 is complete.


## 4. Integral Inequalities of the Hermite–Hadamard Type

In this section, we prove some new inequalities of the Hermite–Hadamard type for co-ordinated MT-s1,s2-convex functions.Theorem 1.
Letf:Δ=a,b×c,d⊆ℝ2⟶ℝ+ for a<b and c<d. If f is coordinated MT-s1,s2-convex on Δ for s1,s2∈0,12 and f∈L1Δ, then(12)122−s1+s2fa+b2,c+d2≤122−s1+s2/21b−a∫abfx,c+d2dx+1d−c∫cdfa+b2,ydy≤1b−ad−c∫cd∫abfx,ydxdy≤12Bs2+2/2,1/22s1+1/2b−a∫abfx,c+fx,ddx+Bs1+2/2,1/22s2+1/2d−c∫cdfa,y+fb,ydy≤Bs1+2/2,1/2Bs2+2/2,1/22s1+s2/2+1fa,c+fb,c+fa,d+fb,d,where Bα,β denotes the well-known beta function which may be defined by(13)Bα,β=∫01tα−11−tβ−1dt,ℛeα,ℛeβ>0.Proof.
For all0<t<1, we have(14)a+b2=ta+1−tb+1−ta+tb2,c+d2=12c+d2+c+d2.
Lettingx=ta+1−tb, z=1−ta+tb, and z=w=c+d/2 in (4) and using the MT-s1,s2-convexity of f, we obtain(15)fa+b2,c+d2≤fta+1−tb,c+d/2+f1−ta+tb,c+d/22s1+s2/2.
Integrating the above inequality overt on 0,1 and utilizing the change of the variable x=ta+1−tb for 0<t<1 result in(16)fa+b2,c+d2≤12s1+s2/2∫01fta+1−tb,c+d2+f1−ta+tb,c+d2dt=12s1+s2/2−1b−a∫abfx,c+d2dx.
By the MT-s1,s2-convexity of f, we obtain(17)1b−a∫abfx,c+d2dx≤12s1+s2/2b−a∫01∫abfx,λc+1−λd+fx,1−λc+λddxdλ=12s1+s2/2−1b−ad−c∫cd∫abfx,ydxdy.
From (16) and (17), it follows that(18)122−s1+s2fa+b2,c+d2≤121−s1+s2/2b−a∫abfx,c+d2dx≤1b−ad−c∫cd∫abfx,ydxdy.
Similarly, we have(19)122−s1+s2fa+b2,c+d2≤121−s1+s2/2d−c∫cdfa+b2,ydy≤1b−ad−c∫cd∫abfx,ydxdy.
A combination of (18) and (19) gives the desired inequality (12).
Puttingy=λc+1−λd for all 0<λ<1 and using the MT-s1,s2-convexity of f reveals(20)1b−ad−c∫cd∫abfx,ydxdy=1b−a∫01∫abfx+x2,λc+1−λddxdλ≤12s1+1/2b−a∫01∫abλs2/21−λfx,c+1−λs2/2λfx,ddxdλ=Bs2+2/2,1/22s1+1/2b−a∫abfx,c+fx,ddx.
Takingx=ta+1−tb for 0<t<1 and employing the MT-s1,s2-convexity of f leads to(21)1b−a∫abfx,cdx=∫01fta+1−tb,c+c2dt≤12s2+1/2∫01ts1/21−tfa,c+1−ts1/2tfb,cdt=Bs1+2/2,1/22s2+1/2fa,c+fb,c,(22)1b−a∫abfx,ddx≤Bs1+2/2,1/22s2+1/2fa,d+fb,d.
Applying inequalities between (19) and (22) arrives at(23)1b−ad−c∫cd∫abfx,ydxdy≤Bs2+2/2,1/22s1+1/2b−a∫abfx,c+fx,ddx≤Bs1+2/2,1/2Bs2+2/2,1/22s1+s2/2+1fa,c+fb,c+fa,d+fb,d.
By similar argument, we can find(24)1b−ad−c∫cd∫abfx,ydxdy≤Bs1+2/2,1/22s2+1/2s1+1Γs1+1/2d−c∫cdfa,y+fb,ydy≤Bs1+2/2,1/2Bs2+2/2,1/22s1+s2/2+1fa,c+fb,c+fa,d+fb,d.
The proof of Theorem1 is complete.Corollary 1.
Under the conditions of Theorem1, if s1=s2=s, then(25)1221−sfa+b2,c+d2≤122−s1b−a∫abfx,c+d2dx+1d−c∫cdfa+b2,ydy≤1b−ad−c∫cd∫abfx,ydxdy≤Bs+2/2,1/22s+1/2+11b−a∫abfx,c+fx,ddx+1d−c∫cdfa,y+fb,ydy≤Bs+2/2,1/222s+1fa,c+fb,c+fa,d+fb,d.Theorem 2.
Letf:Δ=a,b×c,d⊆ℝ2⟶ℝ for a<b and c<d have the second partial derivatives and ∂2f/∂x∂y∈L1Δ and let s1,s2∈0,12. If ∂2f/∂x∂yq for q≥1 is coordinated MT-s1,s2-convex functions on Δ, then(26)If≤b−ad−c41+1/qBs1+2/2,3/2Bs2+2/2,3/241/q∂2fa,c∂x∂yq+s2+2∂2fa,d∂x∂yq+s1+2∂2fb,c∂x∂yq+s1+2s2+2∂2fb,d∂x∂yq1/q,where Bα,β is the Beta function.Proof.
From Lemma1 and Hölder’s integral inequality, it follows that(27)If≤b−ad−c∫01∫011−t1−λdtdλ1−1/q×∫01∫011−t1−λ∂2fta+1−tb,λc+1−λd∂x∂yqdtdλ1/q=b−ad−c41−1/q∫01∫011−t1−λ∂2fta+1−tb,λc+1−λd∂x∂yqdtdλ1/q.
By the coordinated MT-s1,s2-convexity of ∂2f/∂x∂yq, we have(28)∫01∫011−t1−λ∂2fta+1−tb,λc+1−λd∂x∂yqdtdλ≤14∫01∫011−t1−λts1/2λs2/21−t1−λ∂2fa,c∂x∂yq+ts1/21−λs2/21−tλ∂2fa,d∂x∂yq+1−ts1/2λs2/2t1−λ∂2fb,c∂x∂yq+1−ts1/21−λs2/2tλ∂2fb,d∂x∂yqdtdλ=Bs1+2/2,3/2Bs2+2/2,3/24∂2fa,c∂x∂yq+s2+2∂2fa,d∂x∂yq+s1+2∂2fb,c∂x∂yq+s1+2s2+2∂2fb,d∂x∂yq.
Combining (27) and (28) results in (26). Theorem 2 is thus proved.Corollary 2.
Under the assumptions of Theorem2, if s1=s2=s, then(29)If≤b−ad−c41+1/qBs+2/2,3/222/q∂2fa,c∂x∂yq+s+2∂2fa,d∂x∂yq+s+2∂2fb,c∂x∂yq+s+22∂2fb,d∂x∂yq1/q.Theorem 3.
Letf:Δ=a,b×c,d⊆ℝ2⟶ℝ for a<b and c<d have the second partial derivatives and ∂2f/∂x∂y∈L1Δ and let s1,s2∈0,12. If ∂2f/∂x∂yq for q>1 is co-ordinated MT-s1,s2-convex functions on Δ, then(30)If≤b−ad−c41/qq−12q−12−2/qBs1+22,12Bs2+22,121/q×∂2fa,c∂x∂yq+∂2fa,d∂x∂yq+∂2fb,c∂x∂yq+∂2fb,d∂x∂yq1/q,where Bα,β is the Beta function.Proof.
From Lemma1, Hölder’s integral inequality, and the coordinated MT-s1,s2-convexity of ∂2f/∂x∂yq, it follows that(31)If≤b−ad−c∫01∫011−t1−λq/q−1dtdλ1−1/q×∫01∫01∂2fta+1−tb,λc+1−λd∂x∂yqdtdλ1/q=b−ad−cq−12q−12−2/q∫01∫01∂2fta+1−tb,λc+1−λd∂x∂yqdtdλ1/q≤b−ad−c41/qq−12q−12−2/q×∫01∫01ts1/2λs2/21−t1−λ∂2fa,c∂x∂yq+ts1/21−λs2/21−tλ∂2fa,d∂x∂yq+1−ts1/2λs2/2t1−λ∂2fb,c∂x∂yq+1−ts1/21−λs2/2tλ∂2fb,d∂x∂yqdtdλ1/q=b−ad−c41/qq−12q−12−2/qBs1+22,12Bs2+22,121/q×∂2fa,c∂x∂yq+∂2fa,d∂x∂yq+∂2fb,c∂x∂yq+∂2fb,d∂x∂yq1/q.
Theorem3 is thus proved.Corollary 3.
Under the conditions of Theorem3, if s1=s2=s, then(32)If≤b−ad−c41/qq−12q−12−2/qBs+22,122/q×∂2fa,c∂x∂yq+∂2fa,d∂x∂yq+∂2fb,c∂x∂yq+∂2fb,d∂x∂yq1/q.


## 5. Conclusion

In this paper, we conclude the following:(1)
From Definition5, we introduced a new concept of MT-s1,s2-convex functions on the coordinates on the rectangle Δ of the plane ℝ2(2)
From Propositions1 and 2, we investigated simple properties of MT-s1,s2-convex functions on the coordinates on Δ(3)
With the help of the integral identity in Lemma1, via Theorems 1, 2, and 3, and via Corollaries 1, 2, and 3, we established some new Hermite–Hadamard type inequalities for MT-s1,s2-convex functions on the coordinates on Δ



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