In this paper, the authors define the notion of harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex functions, establish a new integral identity, present some new Hermite–Hadamard type integral inequalities for harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex functions, and derive some known results.
Citation: Chun-Ying He, Aying Wan, Bai-Ni Guo. Hermite–Hadamard type inequalities for harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex functions[J]. AIMS Mathematics, 2023, 8(7): 17027-17037. doi: 10.3934/math.2023869
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In this paper, the authors define the notion of harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex functions, establish a new integral identity, present some new Hermite–Hadamard type integral inequalities for harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex functions, and derive some known results.
The following definitions are well known in the literature.
Definition 1.1. A function f:I⊆R=(−∞,∞)→R is said to be convex if
f(λx+(1−λ)y)≤λf(x)+(1−λ)f(y) |
holds for all x,y∈I and λ∈[0,1].
Definition 1.2 ([9]). For b>0 and some fixed number m∈(0,1], let f:[0,b]→R0=[0,∞). If
f(tx+m(1−t)y)≤tf(x)+m(1−t)f(y) |
is valid for all x,y∈[0,b] and t∈[0,1], then we say that f is an m-convex function on [0,b].
Definition 1.3 ([7]). For b>0 and some fixed tuple (α,m)∈(0,1]2, let f:[0,b]→R0. If
f(tx+m(1−t)y)≤tαf(x)+m(1−tα)f(y) |
is valid for all x,y∈[0,b] and t∈[0,1], then we say that f(x) is an (α,m)-convex function on [0,b].
Definition 1.4 ([2,6]). Let s∈(0,1] be a real number. A function f:I⊆R0=[0,∞)→R is said to be s-convex (in the second sense) if
f(λx+(1−λ)y)≤λsf(x)+(1−λ)sf(y) |
holds for all x,y∈I and λ∈[0,1].
Definition 1.5 ([16]). For some number s∈[−1,1], a function f:I⊆R0→R is said to be extended s-convex if
f(λx+(1−λ)y)≤λsf(x)+(1−λ)sf(y) |
is valid for all x,y∈I and λ∈(0,1).
Definition 1.6 ([17]). For some numbers m,α∈(0,1], a function f:(0,b]⊆R+=(0,∞)→R0 is said to be harmonically (α,m)-convex if
f((tx+1−tmy)−1)≤tαf(x)+m(1−tα)f(y) |
is valid for all x,y∈I and t∈(0,1).
Definition 1.7 ([3,4]). A function f:Δ=[a,b]×[c,d]⊆R2→R with a<b and c<d is said to be convex on the coordinates on Δ if the partial mappings
fy:[a,b]→R,fy(u)=fy(u,y) |
and
fx:[c,d]→R,fx(v)=fx(x,v) |
are both convex for x∈(a,b) and y∈(c,d).
A formal definition for coordinated convex functions may be stated as follows.
Definition 1.8 ([3,4]). A function f:Δ→R is said to be convex on the coordinates on Δ=[a,b]×[c,d]⊆R2 with a<b and c<d if the inequality
f(tx+(1−t)z,λy+(1−λ)w)≤tλf(x,y)+t(1−λ)f(x,w)+(1−t)λf(z,y)+(1−t)(1−λ)f(z,w) |
holds for all t,λ∈[0,1],(x,y),(z,w)∈Δ.
Definition 1.9 ([12]). For (s1,m1),(s2,m2)∈[−1,1]×(0,1], a function f:[0,b]×[0,d]→R is said to be extended ((s1,m1)-(s2,m2))-convex on the coordinates on [0,b]×[0,d] if the inequality
f(tx+m1(1−t)z,λy+m2(1−λ)w)≤ts1λs2f(x,y)+m2ts1(1−λ)s2f(x,w)+m1(1−t)s1λs2f(z,y)+m1m2(1−t)s1(1−λ)s2f(z,w) |
holds for all t,λ∈(0,1) and (x,y),(z,w)∈[0,b]×[0,d].
Definition 1.10 ([5]). For f:(0,b]⊆R+→R, m∈(0,1], and s∈[−1,1], the function f is said to be harmonically extended (s,m)-convex on (0,b] if
f((tx+m1−ty)−1)≤tsf(x)+m(1−t)sf(y) |
holds for all x,y∈(0,b] and t∈(0,1).
In a previous paper [3], Dragomir established the following theorem.
Theorem 1.1. ([3, Theorem 1]). Let f:Δ=[a,b]×[c,d]⊆R2→R be convex on the coordinates on Δ. Then
f(a+b2,c+d2)≤12[1b−a∫baf(x,c+d2)dx+1d−c∫dcf(a+b2,y)dy]≤1(b−a)(d−c)∫ba∫dcf(x,y)dydx≤14[1b−a∫ba[f(x,c)+f(x,d)]dx+1d−c∫dc[f(a,y)+f(b,y)]dy]≤14[f(a,c)+f(b,c)+f(a,d)+f(b,d)]. |
There are many other new conclusions in the literature [1,8,10,11,13,14,15,18].
The main purpose of this paper is to introduce the notion of harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex functions and to establish some new Hermite–Hadamard type integral inequalities for this class of convex functions.
We now introduce the concept of harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex functions.
Definition 2.1. For (s1,m1),(s2,m2)∈[−1,1]×(0,1], a function f:Δ=(0,b]×(0,d]⊆R2+→R is said to be harmonic-arithmetic extended (s1,m1)-(s2,m2)-convex on the coordinates on Δ if the inequality
f((tx+m1(1−t)z)−1,(λy+m2(1−λ)w)−1)≤ts1λs2f(x,y)+m2ts1(1−λ)s2f(x,w)+m1(1−t)s1λs2f(z,y)+m1m2(1−t)s1(1−λ)s2f(z,w) |
holds for all t,λ∈(0,1) and (x,y),(z,w)∈Δ.
Example 2.1. Let f(x,y)=1(xy)r for x,y∈R+ and r≥1. For all tuples (s1,m1),(s2,m2)∈[−1,1]×(0,1] and (x,y),(z,w)∈R2+, we have
f((tx+m1(1−t)z)−1,(λy+m2(1−λ)w)−1)≤tzr+(1−t)(m1x)r(xz)rλwr+(1−λ)(m2y)r(yw)r≤(ts1xr+m1(1−t)s1zr)(λs2yr+m2(1−λ)s2wr)=ts1λs2f(x,y)+m2ts1(1−λ)s2f(x,w)+m1(1−t)s1λs1f(z,y)+m1m2(1−t)s1(1−λ)s2f(z,w). |
Therefore, the function f(x,y)=1(xy)r is harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex on R2+.
In order to prove our main results, we need the following lemma.
Lemma 2.1. Let f:Δ⊆R2+→R be a four-time partial differentiable function on Δ=[a,b]×[c,d] with a<b and c<d. If ∂4f∂x2∂y2∈L1(Δ), then
H(f):=f(a,c)+f(b,c)+f(a,d)+f(b,d)4−12(b−a)∫ba[f(x,c)+f(x,d)]dx−12(d−c)∫dc[f(a,y)+f(b,y)]dy+1(b−a)(d−c)∫dc∫baf(x,y)dxdy=(b−a)2(d−c)24(abdc)2∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4(λc+1−λd)−4×∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)dtdλ. |
Proof. Let x=(ta+1−tb)−1 and y=(λc+1−λd)−1 for t,λ∈[0,1]. Integrating by parts, we have
H(f)=(b−a)(d−c)4abdc∫dc∫ba(xy)2[a(b−x)x(b−a)−a2(b−x)2x2(b−a)2][c(d−y)y(d−c)−c2(d−y)2y2(d−c)2]∂4f(x,y)∂x2∂y2dxdy=(b−a)2(d−c)24(abdc)2∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4(λc+1−λd)−4×∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)dtdλ. |
Lemma 2.1 is proved.
In this section, we establish Hermite–Hadamard type integral inequalities for harmonic-arithmetic extended (s1,m1)-(s2,m2) coordinated convex functions.
Theorem 3.1. Let f:Δ=(0,b]×(0,d]⊆R2+→R be a four-time partial differentiable function on Δ such that ∂4f∂x2∂y2∈L1(Δ). If |∂4f∂x2∂y2|q is a harmonic-arithmetic extended (s1,m1)-(s2,m2) convex function on the coordinates on Δ for q≥1 and for some fixed tuples (s1,m1),(s2,m2)∈[−1,1]×(0,1] and (a,c),(b,d)∈Δ with a<b and c<d, then
|H(f)|≤(b−a)2(d−c)2144[(s1+2)(s1+3)(s2+2)(s2+3)]1/q(6acbd)2/q[{S(2,2)|∂4f(a,c)∂x2∂y2|q+m2S(2,s2+2)×|∂4f(a,m2d)∂x2∂y2|q+m1S(s1+2,2)|∂4f(m1b,c)∂x2∂y2|q+m1m2S(s1+2,s2+2)|∂4f(m1b,m2d)∂x2∂y2|q}1/q, | (3.1) |
where
S(u,v)=2F1(4,u,s1+4,b−ab)2F1(4,v,s2+4,d−cd) |
and 2F1(c,d;e;z) is the Gauss hypergeometric function which has the integral representation
2F1(c,d,e;z)=Γ(e)Γ(d)Γ(e−d)∫10td−1(1−t)e−d−1(1−zt)−cdt |
for e>d>0, |z|<1, c∈R, and −1≤u,v≤1 with
Γ(w)=∫∞0tw−1e−tdt,ℜ(w)>0. |
Proof. By Lemma 2.1, by Hölder's integral inequality, and by the coordinated harmonic-arithmetic extended ((s1,m1)-(s2,m2))-convexity of |∂4f∂x2∂y2|q, we have
|H(f)|≤(b−a)2(d−c)24(abdc)2∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4×(λc+1−λd)−4|∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)|dtdλ≤(b−a)2(d−c)24(abdc)2[∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4×(λc+1−λd)−4dtdλ]1−1/q[∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4×(λc+1−λd)−4|∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)|qdtdλ]1/q≤(b−a)2(d−c)24(abdc)2[∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4×(λc+1−λd)−4dtdλ]1−1/q{∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4×(λc+1−λd)−4[ts1λs2|∂4f(a,c)∂x2∂y2|q+m2ts1(1−λ)s2|∂4f(a,m2d)∂x2∂y2|q+m1(1−t)s1λs2|∂4f(m1b,c)∂x2∂y2|q+m1m2(1−t)s1(1−λ)s2|∂4f(m1b,m2d)∂x2∂y2|q]dtdλ}1/q. |
Direct computation gives
∫10t(1−t)(ta+1−tb)−4dt=(ab)26,∫10ts+1(1−t)(ta+1−tb)−4dt=a4(s+2)(s+3)2F1(4,2,s+4,b−ab),∫10t(1−t)s+1(ta+1−tb)−4dt=a4(s+2)(s+3)2F1(4,s+2,s+4,b−ab). |
Combining the last three equalities with the above inequality leads to the inequality (3.1). Theorem 3.1 is proved.
Remark 3.1. Under the conditions of Theorem 3.1, if we put q=1 in Theorem 3.1, then
|H(f)|≤[ac(b−a)(d−c)]24(bd)2(s1+2)(s1+3)(s2+2)(s2+3)[S(2,2)|∂4f(a,c)∂x2∂y2|+m2S(2,s2+2)|∂4f(a,m2d)∂x2∂y2|+m1S(s1+2,2)|∂4f(m1b,c)∂x2∂y2|+m1m2S(s1+2,s2+2)|∂4f(m1b,m2d)∂x2∂y2|]. |
In particular, when s1=s2=s and m1=m2=m, we have
|H(f)|≤[ac(b−a)(d−c)]24(bd)2(s+2)2(s+3)2[S(2,2)|∂4f(a,c)∂x2∂y2|+mS(2,s+2)|∂4f(a,md)∂x2∂y2|+mS(s+2,2)|∂4f(mb,c)∂x2∂y2|+m2S(s+2,s+2)|∂4f(mb,md)∂x2∂y2|]. |
Theorem 3.2. For some fixed tuples (s1,m1),(s2,m2)∈[−1,1]×(0,1], let f:Δ=(0,b]×(0,d]⊆R2+→R be a four-time partial differentiable function on Δ such that ∂4f∂x2∂y2∈L1(Δ). If |∂4f∂x2∂y2|q is a harmonic-arithmetic extended (s1,m1)-(s2,m2) convex function on the coordinates on Δ for q>1 and (a,c),(b,d)∈Δ with a<b and c<d, then
|H(f)|≤14[(b−a)(d−c)abdc]2/q[T(a,b)T(c,d)]1−1/q[(s1+2)(s1+3)(s2+2)(s2+3)]1/q[|∂4f(a,c)∂x2∂y2|q+m2|∂4f(a,m2d)∂x2∂y2|q+m1|∂4f(m1b,c)∂x2∂y2|q+m1m2|∂4f(m1b,m2d)∂x2∂y2|q]1/q, | (3.2) |
where
T(a,b)=(q−1)2{[(q+3)b−(3q+1)a]b2(q+1)/(q−1)+[(3q+1)b−(q+3)a]a2(q+1)/(q−1)}2(q+1)(q+3)(3q+1)(b−a). |
Proof. Using Lemma 2.1 and Hölder's integral inequality, we have
|H(f)|≤(b−a)2(d−c)24(abdc)2∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4×(λc+1−λd)−4|∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)|dtdλ≤(b−a)2(d−c)24(abdc)2[∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4q/(q−1)×(λc+1−λd)−4q/(q−1)dtdλ]1−1/q[∫10∫10tλ(1−t)(1−λ)×|∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)|qdtdλ]1/q |
and, by the coordinated harmonic-arithmetic extended ((s1,m1)-(s2,m2))-convexity of |∂4f∂x2∂y2|q,
∫10∫10tλ(1−t)(1−λ)|∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)|qdtdλ≤∫10∫10tλ(1−t)(1−λ)[ts1λs2|∂4f(a,c)∂x2∂y2|q+m2ts1(1−λ)s2|∂4f(a,m2d)∂x2∂y2|q+m1(1−t)s1λs2|∂4f(m1b,c)∂x2∂y2|q+m1m2(1−t)s1(1−λ)s2|∂4f(m1b,m2d)∂x2∂y2|q]dtdλ. |
Furthermore, a straightforward computation gives
∫10t(1−t)(ta+1−tb)−4q/(q−1)dt=(ab)2(b−a)2T(a,b) |
and, for s≥−1,
∫10ts+1(1−t)dt=1(s+2)(s+3). |
Combining the last two equalities and the above inequalities gives the desired result (3.2). The proof of Theorem 3.2 is complete.
Remark 3.2. Under the conditions of Theorem 3.2, if s1=s2=s and m1=m2=m, we have
|H(f)|≤[(b−a)(d−c)2qabdc]2/q[T(a,b)T(c,d)]1−1/q[(s+2)(s+3)]2/q{|∂4f(a,c)∂x2∂y2|q+m|∂4f(a,md)∂x2∂y2|q+m|∂4f(mb,c)∂x2∂y2|q+m2|∂4f(mb,md)∂x2∂y2|q}1/q. |
Theorem 3.3. Under the conditions of Theorem 3.2, we have
|H(f)|≤14[(b−a)(d−c)abdc]1/q+1[(q−1)2(3q+1)2(b(3q+1)/(q−1)−a(3q+1)/(q−1))(d(3q+1)/(q−1)−c(3q+1)/(q−1))]1−1/q×[R(s1,0,s2,0)|∂4f(a,c)∂x2∂y2|q+m2R(s1,0,0,s2)|∂4f(a,m2d)∂x2∂y2|q+m1R(0,s1,s2,0)|∂4f(m1b,c)∂x2∂y2|q+m1m2R(0,s1,0,s2)|∂4f(m1b,m2d)∂x2∂y2|q]1/q, |
where
R(u,v,e,ℓ)=B(u+q+1,v+q+1)B(e+q+1,ℓ+q+1) |
for u,v,e,ℓ≥0 and B(x,y) is the beta function, which can be defined by
B(x,y)=∫10tx−1(1−t)y−1dt,x,y>0. |
Proof. Using Lemma 2.1 and Hölder's integral inequality, we have
|H(f)|≤(b−a)2(d−c)24(abdc)2∫10∫10tλ(1−t)(1−λ)(ta+1−tb)−4×(λc+1−λd)−4|∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)|dtdλ≤(b−a)2(d−c)24(abdc)2[∫10∫10(ta+1−tb)−4q/(q−1)(λc+1−λd)−4q/(q−1)dtdλ]1−1/q×[∫10∫10[tλ(1−t)(1−λ)]q|∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)|qdtdλ]1/q, |
where
∫10(ta+1−tb)−4q/(q−1)dt=(q−1)ab(3q+1)(b−a)(b(3q+1)/(q−1)−a(3q+1)/(q−1)). |
By the coordinated harmonic-arithmetic extended ((s1,m1)-(s2,m2))-convexity of |∂4f∂x2∂y2|q, we obtain
∫10∫10[tλ(1−t)(1−λ)]q|∂4∂t2∂λ2f((ta+1−tb)−1,(λc+1−λd)−1)|qdtdλ≤∫10∫10[tλ(1−t)(1−λ)]q[ts1λs2|∂4f(a,c)∂x2∂y2|q+m2ts1(1−λ)s2|∂4f(a,m2d)∂x2∂y2|q+m1(1−t)s1λs2|∂4f(m1b,c)∂x2∂y2|q+m1m2(1−t)s1(1−λ)s2|∂4f(m1b,m2d)∂x2∂y2|q]dtdλ=R(s1,0,s2,0)|∂4f(a,c)∂x2∂y2|q+m2R(s1,0,0,s2)|∂4f(a,m2d)∂x2∂y2|q+m1R(0,s1,s2,0)|∂4f(m1b,c)∂x2∂y2|q+m1m2R(0,s1,0,s2)|∂4f(m1b,m2d)∂x2∂y2|q. |
Combining the above equality and the above two inequalities, we complete the proof of Theorem 3.3.
Remark 3.3. Under the conditions of Theorem 3.3, if s1=s2=s and m1=m2=m, then
|H(f)|≤14[(b−a)(d−c)abdc]1/q+1[(q−1)2(3q+1)2(b(3q+1)/(q−1)−a(3q+1)/(q−1))(d(3q+1)/(q−1)−c(3q+1)/(q−1))]1−1/q×[R(s,0,s,0)|∂4f(a,c)∂x2∂y2|q+mR(s,0,0,s)|∂4f(a,md)∂x2∂y2|q+mR(0,s,s,0)|∂4∂x2∂y2f(mb,c)|q+m2R(0,s,0,s)|∂4f(mb,md)∂x2∂y2|q]1/q. |
Theorem 3.4. Let f:Δ=(0,b]×(0,d]⊆R2+→R and f∈L1(Δ). Denote H(a,b)=2aba+b. If f is a harmonic-arithmetic extended (s1,m1)-(s2,m2)-convex function on the coordinates on Δ for some fixed tuples (s1,m1),(s2,m2)∈(−1,1]×(0,1] and (a,c),(b,d)∈Δ with a<b and c<d, then
f(H(a,b),H(c,d))≤12s1+s2abcd(b−a)(d−c)∫dc∫baf(x,y)+m1f(m1x,y)+m2f(x,m2y)+m1m2f(m1x,m2y)x2y2dxdy |
and
∫dc∫baf(x,y)x2y2dxdy≤(b−a)(d−c)abcdf(a,c)+m2f(a,m2d)+m1f(m1b,c)+m1m2f(m1b,m2d)(s1+1)(s2+1). |
Proof. Since
H(a,b)=2ta+1−tb+1−ta+tb |
for t∈[0,1], by the coordinated harmonic-arithmetic extended ((s1,m1)-(s2,m2))-convexity of f, we have
f(H(a,b),H(c,d))≤12s1+s2∫10∫10[f((ta+1−tb)−1,(λc+1−λd)−1)+m1f(m1(1−ta+tb)−1,(λc+1−λd)−1)+m2f((ta+1−tb)−1,m2(1−λc+λd)−1)+m1m2f(m1(1−ta+tb)−1,m2(1−λc+λd)−1)]dtdλ. |
Setting x=(ta+1−tb)−1 and y=(λc+1−λd)−1 for t,λ∈(0,1), we have
∫10∫10f((ta+1−tb)−1,(λc+1−λd)−1)dtdλ=abcd(b−a)(d−c)∫dc∫baf(x,y)x2y2dxdy. |
Combining the above equality and inequality yields the first inequality in Theorem 3.4.
On the other hand, we have
abcd(b−a)(d−c)∫dc∫baf(x,y)x2y2dxdy=∫10∫10f((ta+1−tb)−1,(λc+1−λd)−1)dtdλ≤∫10∫10[ts1λs2f(a,c)+m2ts1(1−λ)s2f(a,m2d)+m1(1−t)s1λs2f(m1b,c)+m1m2(1−ts1)(1−λ)s2f(m1b,m2d)]dtdλ=f(a,c)+m2f(a,m2d)+m1f(m1b,c)+m1m2f(m1b,m2d)(s1+1)(s2+1). |
The second inequality in Theorem 3.4 is proved.
Corollary 3.1. Under the conditions of Theorem 3.4, if m1=m2=1, then
f(H(a,b),H(c,d))≤12s1+s2−2abcd(b−a)(d−c)∫dc∫baf(x,y)x2y2dxdy≤f(a,c)+f(a,d)+f(b,c)+f(b,d)2s1+s2−2(s1+1)(s2+1). |
Next, we give an application.
Theorem 3.5. Let b>a>0,d>c>0, and r≥1. Then
[(a+b)(c+d)4]r≤(br+1−ar+1)(dr+1−cr+1)(r+1)2(b−a)(d−c)≤(ar+br)(cr+dr)4. |
In particular, when r=2, we have
(a+b)2(c+d)216≤(a2+ab+b2)(c2+cd+d2)9≤(a2+b2)(c2+d2)4. |
Proof. Using Example 2.1 and Corollary 3.1, we obtain
[(a+b)(c+d)4abcd]r≤abcd(b−r−1−a−r−1)(d−r−1−c−r−1)(r+1)2(b−a)(d−c)≤(ac)−r+(ad)−r+(bc)−r+(bd)−r4. |
After simplification, Theorem 3.5 is proved.
This work was partially supported by the National Natural Science Foundation of China (Grant No. 12061033) and by the Science Research Fund of Hulunbuir University (Grant No. 2022ZKYB06), China.
The authors appreciate anonymous referees for their valuable comments, careful corrections, and helpful suggestions to the original version of this paper.
The authors declare that they have no conflict of interest.
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