Int. J. Anal. Appl. (2022), 20:36

MϕA-h-Convexity and Hermite-Hadamard Type Inequalities

Sanja Varošanec∗

Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička 30, 10000 Zagreb,

Croatia

∗Corresponding author: varosans@math.hr

Abstract. We investigate a family of MϕA-h-convex functions, give some properties of it and several

inequalities which are counterparts to the classical inequalities such as the Jensen inequality and the

Schur inequality. We give the weighted Hermite-Hadamard inequalities for an MϕA-h-convex function

and several estimations for the product of two functions.

1. Preliminaries

It is known that the classical convexity can be generalized to an MN-convexity, where M and N

are means which is described in [8]. The other direction of generalization leads to the concept of

h-convexity, [13]. It is interesting to see properties of a function which definition combines some

elements of MN-convexity and of h-convexity.

Let M and N be two means in two variables. We say that a function f : I →R is MN-convex if

f (M(x,y))≤ N(f (x), f (y))

for every x,y ∈ I.
In this paper we will focus on a somewhat special type of means.

Let ϕ be a continuous, strictly monotone function defined on the interval I. By Mϕ we denote a

quasi-arithmetic mean:

Mϕ(x,y;t,1− t) := ϕ−1(tϕ(x)+(1− t)ϕ(y)), x,y ∈ I,t ∈ [0,1].

It is obvious that the power mean Mp corresponds to ϕ(x)= xp if p 6=0 and to ϕ(x)= logx if p =0.

Received: Jun. 26, 2022.

2010 Mathematics Subject Classification. 26A51, 26D15.
Key words and phrases. the Hermite-Hadamard inequality; the Jensen inequality; MϕA-h-convex function; quasi-

arithmetic mean; the Schur inequality.

https://doi.org/10.28924/2291-8639-20-2022-36
ISSN: 2291-8639

© 2022 the author(s).

https://doi.org/10.28924/2291-8639-20-2022-36


2 Int. J. Anal. Appl. (2022), 20:36

Let ϕ and ψ be two continuous, strictly monotone functions defined on intervals I and K respectively.

Let h : J → R be a non-negative function, (0,1) ⊆ J and let f : I → K such that h(t)ψ(f (x))+
h(1− t)ψ(f (y))∈ ψ(K) for all x,y ∈ I,t ∈ (0,1). We say that a function f is MϕMψ-h-convex if

f (Mϕ(x,y;t,1− t))≤ Mψ(f (x), f (y);h(t),h(1− t))

for all x,y ∈ I and all t ∈ (0,1). Especially, a function f : I →R is called MϕA-h-convex if

f (Mϕ(x,y;t,1− t))≤ h(t)f (x)+h(1− t)f (y) (1.1)

for all x,y ∈ I and t ∈ (0,1). The MϕMψ-h-concavity is defined on a natural way.
Some particular cases of MϕMψ-h-convex functions are recently investigated. If h(t) = t, then

the MϕMψ-h-convexity collapses to the MϕMψ-convexity which is described in [8]. If Mϕ, Mψ are an

arithmetic mean (A), a geometric mean (G) or a harmonic mean (H), then we can find several results.

For example, AA-h-convexity or simply h-convexity firstly appeared in [13]. An HA-h-convexity or

harmonic-h-convexity is described in [2] and [10]. HG-h-convexity investigated in [10] and AG-h-

convexity or log-h-convexity in [9]. AMp-h-convexity or (h,p)-convexity is described in [6] while some

properties of MpA-h-convex functions are given in [4]. Also, we have to mention article [1] devoted

to the MN-h-convexity where M,N ∈{A,G,H}.
The aim of this paper is to give several statements about MϕA-h-convex functions primarly related

to the Hermite-Hadamard inequality and the Jensen inequality. The following section is devoted to

the properties of MϕA-h-convex functions. Also in that section we give counterparts to the Jensen

and the Schur inequality and some related results. In the third section we prove several inequalities of

Hermite-Hadamard type.

2. Properties of MϕA-h-convex functions and Jensen-type inequalities

Proposition 2.1. Let ϕ be a continuous, strictly monotone function defined on the interval I. Let

h be a non-negative function defined on the interval J, (0,1) ⊆ J. A function f is MϕA-h-convex
(concave) on I if and only if the function f ◦ϕ−1 is h-convex (concave) on ϕ(I).

Proof. Let us suppose thatf is MϕA-h-convex on I and let u,v ∈ ϕ(I), t ∈ (0,1). Since ϕ is
continuous and strictly monotone on I, there exist x,y ∈ I such that u = ϕ(x),v = ϕ(y). Then

(f ◦ϕ−1)(tu +(1− t)v) = (f ◦ϕ−1)(tϕ(x)+(1− t)ϕ(y)))

= f (Mϕ(x,y;t,1− t))≤ h(t)f (x)+h(1− t)f (y)

= h(t)f (ϕ−1(u))+h(1− t)f (ϕ−1(v))

= h(t)(f ◦ϕ−1)(u)+h(1− t)(f ◦ϕ−1)(v)

which means that f ◦ϕ−1 is h-convex. The second case is proved similarly. �



Int. J. Anal. Appl. (2022), 20:36 3

Proposition 2.2. Let ϕ be a continuous, strictly monotone function defined on the interval I. Let

h,h1,h2 be non-negative functions defined on the interval J, (0,1)⊆ J.
(i) Let h1 and h2 have a property

h2(t)≤ h1(t), t ∈ (0,1).

If f : I → [0,∞) is MϕA-h2-convex, then f is an MϕA-h1-convex function.
(ii) If f ,g are MϕA-h-convex functions, λ > 0, then f +g and λf are MϕA-h-convex.

(iii) Let f , : I → [0,∞) be similarly ordered functions on I, i.e.

(f (x)− f (y))(g(x)−g(y))≥ 0, x,y ∈ I

and h(t)+h(1− t)≤ c for all t ∈ (0,1), where h =max{h1,h2} and c is a fixed positive number. If
f is MϕA-h1-convex and g is MϕA-h2-convex, then the product f g is MϕA-h-convex.

Proof. The proof is based on the known results for h-convex functions and characterization given in

Proposition 2.1. Let us prove part (i). If f is MϕA-h2-convex, then f ◦ϕ−1 is h2-convex. Then, using
Proposition 8 from [13], we get that f ◦ϕ−1 is h1-convex, i.e. f is MϕA-h1-convex.

Other parts are proved similarly by applying Propositions 9 and 10 from [13]. �

The following theorem gives a counterpart of the Schur inequality.

Theorem 2.1. Let h be a non-negative supermultiplicative function defined on the interval J, (0,1)⊆
J. Let ϕ be a continuous, strictly monotone function defined on the interval I. Let f : I → [0,∞) be
MϕA-h-convex.

If ϕ is increasing, then for any x1,x2,x3 ∈ I such that x1 < x2 < x3 and ϕ(x3)−ϕ(x2), ϕ(x3)−ϕ(x1),
ϕ(x2)−ϕ(x1)∈ J the following holds

h(ϕ(x3)−ϕ(x2))f (x1)−h(ϕ(x3)−ϕ(x1))f (x2)+h(ϕ(x2)−ϕ(x1))f (x3)≥ 0. (2.1)

If ϕ is decreasing, then for any x1,x2,x3 ∈ I such that x1 < x2 < x3 and ϕ(x2)−ϕ(x3), ϕ(x1)−ϕ(x3),
ϕ(x1)−ϕ(x2)∈ J the following holds

h(ϕ(x2)−ϕ(x3))f (x1)−h(ϕ(x1)−ϕ(x3))f (x2)+h(ϕ(x1)−ϕ(x2))f (x3)≥ 0. (2.2)

Proof. Let assume that ϕ is increasing. For x1,x2,x3 ∈ I such that x1 < x2 < x3 we have

u1 := ϕ(xi) < u2 := ϕ(x2) < u3 := ϕ(x3).

Since a function g := f ◦ϕ−1 is h-convex, using Proposition 16 from [13], we get

h(u3 −u2)g(u1)−h(u3 −u1)g(u2)+h(u2 −u1)g(u3)≥ 0

and after appropriate substitutions we obtain inequality (2.1). Inequality (2.2) is proved in a similar

way. �



4 Int. J. Anal. Appl. (2022), 20:36

The following theorem is a counterpart of the discrete Jensen inequality and its converse for an

MϕA-h-convex function.

Theorem 2.2. Let h : J → R be a non-negative supermultiplicative function, (0,1) ⊆ J. Let ϕ be
a continuous, strictly monotone function defined on the interval I. Let f : I → [0,∞) be a MϕA-
h-convex function. Let w1, . . . ,wn be non-negative real numbers such that Wn =

∑n
i=1wi 6= 0 and

wi
Wn
∈ J, i =1, . . . ,n.

(i) Then for all x1, . . . ,xn ∈ I the following holds

f

(
ϕ−1

(
1

Wn

n∑
i=1

wiϕ(xi)

))
≤

n∑
i=1

h

(
wi
Wn

)
f (xi).

(ii) Then for all x1, . . . ,xn ∈ (a,b)⊆ I the following holds

n∑
i=1

h

(
wi
Wn

)
f (xi) ≤ f (a)

n∑
i=1

h

(
wi
Wn

)
h

(
ϕ(b)−ϕ(xi)
ϕ(b)−ϕ(a)

)

+f (b)

n∑
i=1

h

(
wi
Wn

)
h

(
ϕ(xi)−ϕ(a)
ϕ(b)−ϕ(a)

)
.

Proof. Since f is a MϕA-h-convex function, then f ◦ϕ−1 is h-convex on ϕ(I) and using the Jensen
inequality for h-convex functions and its converse ( [13, Theorems 19 and 21]), we get the above

results. �

The following result is a property of subadditivity for an index set function. Let K be a finite

non-empty set of positive integers. Let us define the index set function F by

F(K)= h(WK)f

(
ϕ−1

(
1

WK

∑
i∈K

wiϕ(xi)

))
−
∑
i∈K

h(wi)f (xi),

where wi ∈ J, WK :=
∑
i∈K wi ∈ J, xi ∈ I.

Theorem 2.3. Let h : J → R be a non-negative supermultiplicative function and let M and K be
finite non-empty sets of positive integers with M ∩ K = ∅. Let wi > 0, (i ∈ M ∪ K) be such that
WK,WM,WM∪K ∈ J. Let ϕ be a continuous, strictly monotone function defined on the interval I.

If f : I → [0,∞) is MϕA-h-convex, then the following inequality holds

F(M ∪K)≤ F(M)+F(K).

Furthermore, if Mk := {1, . . . ,k}, k =2, . . . ,n and WMk ∈ J, then

F(Mn)≤ F(Mn−1)≤ . . . ≤ F(M2)≤ 0



Int. J. Anal. Appl. (2022), 20:36 5

and

F(Mn) ≤ min
1≤i<j≤n

{
h(wi +wj)f

(
ϕ−1

(
wiϕ(xi)+wjϕ(xj)

wi +wj

))
−h(wi)f (xi)−h(wj)f (xj)

}
.

Proof. Let us consider the following difference

F(M)+F(K)−F(M ∪K)

= h(WM)f

(
ϕ−1

(
1

WM

∑
i∈M

wiϕ(xi)

))
+h(WK)f

(
ϕ−1

(
1

WK

∑
i∈K

wiϕ(xi)

))

−h(WM∪K)f

(
ϕ−1

(
1

WM∪K

∑
i∈M∪K

wiϕ(xi)

))
.

Since numbers u := 1
WM

∑
i∈M wiϕ(xi) and v :=

1
WK

∑
i∈K wiϕ(xi) belong to ϕ(I), there exist numbers

x,y ∈ I such that ϕ(x) = u, ϕ(y) = v. Using a definition of the MϕA-h-convexity for t = WMWM∪K ,
1− t = WK

WM∪K
, and x,y and supermultiplicativity of h, we get

f

(
Mϕ(x,y;

WM
WM∪K

,
WK
WM∪K

)

)
≤

h(WM)

h(WM∪K)
f (x)+

h(WK)

h(WM∪K)
f (y) (2.3)

and inequality F(M)+F(K)−F(M ∪K)≥ 0 follows from (2.3) immediately. �

Remark 2.1. If Mϕ = A, then the above results related to the Jensen inequality, its converse and

to the index set function for an h-function were proved in [13].

If Mϕ = H, then the Jensen type inequality for HA-h-convex function is given in [2]. If Mϕ = Mp
and h(t)= t, then the Jensen inequality for MpA-convex was proved in [4]. If Mϕ ∈{A,G,H}, then
results from this section are given in [1].

3. Hermite-Hadamard type inequality and related results

Counterparts of the Hermite-Hadamard inequality appear in the study of every kind of convexity.

Namely, in the classical convexity, the left-hand side or the right-hand side of the Hermite-Hadamard

inequality are equivalent to the definition of convexity. The Hermite-Hadamard inequality for an h-

convex function was proved in [3] and [11] and has the following form.

If h is an integrable function, h(1
2
) 6= 0, then for an integrable h-convex function f : [a,b] → R,

the following sequence of inequalities hold:

1

2h(1
2
)
f

(
a+b

2

)
≤

1

b−a

∫ b
a

f (x)dx ≤ [f (a)+ f (b)]
∫ 1
0

h(x)dx. (3.1)

This section begins with the weighted Hermite-Hadamard inequality for an MϕA-h-convex function.

This result is usually called the Hermite-Hadamard-Féjer inequality.



6 Int. J. Anal. Appl. (2022), 20:36

Theorem 3.1. Let h be a non-negative function defined on the interval J, (0,1) ⊆ J, h(1
2
) 6= 0 and

ϕ be a differentiable, strictly monotone function defined on [a,b].

Let w : [a,b]→ [0,∞) be a function such that wϕ′ ∈ L([a,b]) and

w
(
ϕ−1(tϕ(a)+(1− t)ϕ(b))

)
= w

(
ϕ−1((1− t)ϕ(a)+ tϕ(b))

)
(3.2)

for all t ∈ (0,1). If f is MϕA-h-convex, f wϕ′ ∈ L([a,b]), then

1

2h(1
2
)
f

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))∫ b
a

w(x)ϕ′(x)dx

≤
∫ b
a

f (x)w(x)ϕ′(x)dx (3.3)

≤ [f (a)+ f (b)]
∫ b
a

h

(
ϕ(b)−ϕ(x)
ϕ(b)−ϕ(a)

)
w(x)ϕ′(x)dx,

provided that all integrals exist. Moreover,

1

2h(1
2
)
f

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))
≤

1

ϕ(b)−ϕ(a)

∫ b
a

f (x)ϕ′(x)dx

≤ [f (a)+ f (b)]
∫ 1
0

h(x)dx, (3.4)

provided that all integrals exist.

Proof. Let us prove the first inequality in (3.3). Since ϕ is continuous, strictly monotone, then for fixed

t ∈ (0,1) there exist u,v ∈ [a,b] such that ϕ(u)= tϕ(a)+(1−t)ϕ(b)and ϕ(v)= (1−t)ϕ(a)+tϕ(b).
Then, we get

f

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))
= f

(
ϕ−1

(
1

2
[tϕ(a)+(1− t)ϕ(b)]+

1

2
[(1− t)ϕ(a)+ tϕ(b)]

))
≤ h

(
1

2

)
f (u)+h

(
1

2

)
f (v).

Multiplying the above inequality with w
(
ϕ−1(tϕ(a)+(1− t)ϕ(b))

)
, integrating over [0,1] and using

condition (3.2), we get

f

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))∫ 1
0

w
(
ϕ−1(tϕ(a)+(1− t)ϕ(b))

)
dt

≤ h
(
1

2

)∫ 1
0

f (ϕ−1(tϕ(a)+(1− t)ϕ(b)))w
(
ϕ−1(tϕ(a)+(1− t)ϕ(b))

)
dt

+h

(
1

2

)∫ 1
0

f (ϕ−1((1− t)ϕ(a)+ tϕ(b)))w
(
ϕ−1((1− t)ϕ(a)+ tϕ(b))

)
dt

=
2h(1

2
)

ϕ(b)−ϕ(a)

∫ b
a

f (x)w(x)ϕ′(x)dx

and the first inequality in (3.3) is proved.



Int. J. Anal. Appl. (2022), 20:36 7

Multiplying inequality (1.1) with w
(
ϕ−1(tϕ(a)+(1− t)ϕ(b))

)
and integrating, we get∫ 1

0

f (ϕ−1(tϕ(a)+(1− t)ϕ(b)))wϕ−1(tϕ(a)+(1− t)ϕ(b))dt

≤
∫ 1
0

[h(t)f (a)+h(1− t)f (b)]wϕ−1(tϕ(a)+(1− t)ϕ(b))dt.

Using condition (3.2) and substitution tϕ(a)+(1− t)ϕ(b) = ϕ(x), we get the second inequality in
(3.3).

The last inequality follows from (3.3) for the particular weight w(t)=1. �

Remark 3.1. Some particular results related to the non-weighted Hermite-Hadamard inequality are

know. If h(t) = t, then the counterpart of the Hermite-Hadamard inequality (3.4) is given in [7].

The Hermite-Hadamard inequality (3.4) for a HA-h-convex function is given in [10], see also [15].

Inequality (3.4) for an MϕA-convex function is given in [14] and for MpA-h-convex is given in [5].

Corollary 3.1. Let h be a non-negative function defined on the interval J, (0,1)⊆ J. Let w : [a,b]→
[0,∞), [a,b]⊂ (0,∞) be a function such that

w(atb1−t)= w(a1−tbt)

for all t ∈ (0,1). If f is GA-h-convex, then

1

2h(1
2
)
f (
√
ab)

∫ b
a

w(x)

x
dx ≤

∫ b
a

f (x)
w(x)

x
dx

≤ [f (a)+ f (b)]
∫ b
a

h

(
logb/x

logb/a

)
w(x)

x
dx,

provided that all integrals exist. Furthermore,

1

2h(1
2
)
f (
√
ab)≤

∫ b
a

f (x)

x
dx ≤ [f (a)+ f (b)]

∫ 1
0

h(t)dt,

provided that all integrals exist.

Proof. Putting in inequalities (3.3) and (3.4) ϕ(x)= logx, we get the required results. �

The following theorem contains estimations for the integral mean of the product of two MϕA-h-

convex functions.

Theorem 3.2. Let ϕ be a differentiable, strictly monotone function defined on the interval [a,b]. Let

hi, i = 1,2 be non-negative functions defined on the interval Ji, (0,1) ⊆ Ji, and let f ,g : [a,b] →
[0,∞).

If f is MϕA-h1-convex and g is MϕA-h2-convex, then the following hold:



8 Int. J. Anal. Appl. (2022), 20:36

(i)

1

ϕ(b)−ϕ(a)

∫ b
a

f (x)g(x)ϕ′(x)dx

≤ M(a,b)
∫ 1
0

h1(t)h2(t)dt +N(a,b)

∫ 1
0

h1(t)h2(1− t)dt (3.5)

(ii)

1

2h1(
1
2
)h2(

1
2
)
f

(
Mϕ(a,b;

1

2
,
1

2
)

)
g

(
Mϕ(a,b;

1

2
,
1

2
)

)
−

1

ϕ(b)−ϕ(a)

∫ b
a

f (x)g(x)ϕ′(x)dx

≤ M(a,b)
∫ 1
0

h1(t)h2(1− t)dt +N(a,b)
∫ 1
0

h1(t)h2(t)dt, (3.6)

where h1(
1
2
)h2(

1
2
) 6=0.

(iii)

1

2(ϕ(b)−ϕ(a))2

∫ b
a

∫ b
a

∫ 1
0

ϕ′(x)ϕ′(y)f (Mϕ(x,y;t,1− t))g(Mϕ(x,y;t,1− t))dt dy dx

≤
1

ϕ(b)−ϕ(a)

∫ b
a

f (x)g(x)ϕ′(x)dx

∫ 1
0

h1(t)h2(t)dt

+[M(a,b)+N(a,b)]

∫ 1
0

h1(t)dt

∫ 1
0

h2(t)dt

∫ 1
0

h1(t)h2(1− t)dt (3.7)

(iv)

1

ϕ(b)−ϕ(a)

∫ b
a

∫ 1
0

ϕ′(x)f (Mϕ(x,ϕ
−1
(
ϕ(a)+ϕ(b)

2

)
;t,1− t))×

×g(Mϕ(x,ϕ−1
(
ϕ(a)+ϕ(b)

2

)
;t,1− t))dt dx

≤
1

ϕ(b)−ϕ(a)

∫ b
a

f (x)g(x)ϕ′(x)dx

∫ 1
0

h1(t)h2(t)dt

+[M(a,b)+N(a,b)]

{
h1

(
1

2

)
h2

(
1

2

)∫ 1
0

h1(t)h2(t)dt

+

[
h1

(
1

2

)∫ 1
0

h2(t)dt +h2

(
1

2

)∫ 1
0

h1(t)dt

]∫ 1
0

h1(t)h2(1− t)dt
}
, (3.8)

where

M(a,b)= f (a)g(a)+ f (b)g(b), N(a,b)= f (a)g(b)+ f (b)g(a)

and provided that all integrals exist.

Proof. (i) Since f is MϕA-h1-convex and g is MϕA-h2-convex, we get

f (Mϕ((a,b;t,1−t))≤ h1(t)f (a)+h1(1−t)f (b) and g(Mϕ(a,b;t,1−t))≤ h2(t)g(a)+h2(1−t)g(b).



Int. J. Anal. Appl. (2022), 20:36 9

Multiplying these two inequalities and integrating it over [0,1], we obtain∫ 1
0

f (Mϕ(a,b;t,1− t))g(Mϕ(a,b;t,1− t))dt

≤ M(a,b)
∫ 1
0

h1(t)h2(t)dt +N(a,b)

∫ 1
0

h1(t)h2(1− t)dt

and after a substitution Mϕ(a,b;t,1− t)= x, we get inequality (3.5).
(ii) Since

ϕ(a)+ϕ(b)

2
=
1

2
(tϕ(a)+(1− t)ϕ(b))+

1

2
((1− t)ϕ(a)+ tϕ(b))

for t ∈ (0,1) and since f is MϕA-h1-convex and g is MϕA-h2-convex, we get

f

(
Mϕ(u,v;

1

2
,
1

2
)

)
≤ h1

(
1

2

)
[f (u)+ f (v)]

and

g

(
Mϕ(u,v;

1

2
,
1

2
)

)
≤ h2

(
1

2

)
[g(u)+g(v)],

where ϕ(u) = tϕ(a)+(1− t)ϕ(b) and ϕ(v) = (1− t)ϕ(a)+ tϕ(b). Multiplying these inequalities,
we obtain

f

(
Mϕ(a,b;

1

2
,
1

2
)

)
g

(
Mϕ(a,b;

1

2
,
1

2
)

)
= f

(
Mϕ(u,v;

1

2
,
1

2
)

)
g

(
Mϕ(u,v;

1

2
,
1

2
)

)
≤ h1

(
1

2

)
h2

(
1

2

){
f (u)g(u)+ f (v)g(v)+ f (u)g(v)+ f (v)g(u)

}
≤ h1

(
1

2

)
h2

(
1

2

){
[f (u)g(u)+ f (v)g(v)]+ f (a)g(a)[h1(t)h2(1− t)+h1(1− t)h2(t)]

+f (a)g(b)[h1(t)h2(t)+h1(1− t)h2(1− t)]+ f (b)g(a)[h1(1− t)h2(1− t)+h1(t)h2(t)]

+f (b)g(b)[h1(1− t)h2(t)+h1(t)h2(1− t)]
}
,

where in the last inequality we used the MϕA-h-convexity again. Integrating the above inequality and

using into account that∫ 1
0

f (u)g(u)dt =

∫ 1
0

f (v)g(v)dt =
1

ϕ(b)−ϕ(a)

∫ b
a

f (x)g(x)ϕ′(x)dx

∫ 1
0

h1(t)h2(1− t)dt =
∫ 1
0

h1(1− t)h2(t)dt,
∫ 1
0

h1(t)h2(t)dt =

∫ 1
0

h1(1− t)h2(1− t)dt

we obtain inequality (3.6).

(iii) Since f is MϕA-h1-convex and g is MϕA-h2-convex, we get

f (Mϕ(x,y;t,1− t))≤ h1(t)f (x)+h1(1− t)f (y)

and

g(Mϕ(x,y;t,1− t))≤ h2(t)g(x)+h2(1− t)g(y).



10 Int. J. Anal. Appl. (2022), 20:36

Multiplying these two inequalities, then multiplying with ϕ′(x)ϕ′(y) and integrating it over [a,b] with

respect to x and y and over [0,1] with respect to t, we obtain∫ b
a

∫ b
a

∫ 1
0

ϕ′(x)ϕ′(y)f (Mϕ(x,y;t,1− t))g(Mϕ(x,y;t,1− t))dt dy dx

≤ 2(ϕ(b)−ϕ(a))
∫ 1
0

h1(t)h2(t)dt

∫ b
a

f (x)g(x)ϕ′(x)dx

+2

∫ 1
0

h1(1− t)h2(t)dt
∫ b
a

f (x)ϕ′(x)dx

∫ b
a

g(x)ϕ′(x)dx.

Using the right-hand side of inequality (3.4) to estimate
∫b
a
f (x)ϕ′(x)dx and

∫b
a
g(x)ϕ′(x)dx and

some simple transformations, we get (3.7).

(iv) In this case we begin with inequalities

f (Mϕ(x,ϕ
−1
(
ϕ(a)+ϕ(b)

2

)
;t,1− t))≤ h1(t)f (x)+h1(1− t)f

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))

g(Mϕ(x,ϕ
−1
(
ϕ(a)+ϕ(b)

2

)
;t,1− t))≤ h2(t)g(x)+h2(1− t)g

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))
and proceed in the similar way, i.e. multiply them mutually, then multiply with ϕ′(x) and integrate

with respect to x and t. We get∫ b
a

∫ 1
0

ϕ′(x)f (Mϕ(x,ϕ
−1
(
ϕ(a)+ϕ(b)

2

)
;t,1− t))×

×g(Mϕ(x,ϕ−1
(
ϕ(a)+ϕ(b)

2

)
;t,1− t))dt dx

≤
∫ b
a

f (x)g(x)ϕ′(x)dx

∫ 1
0

h1(t)h2(t)dt

+g

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))∫ b
a

f (x)ϕ′(x)dx

∫ 1
0

h1(t)h2(1− t)dt

+f

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))∫ b
a

g(x)ϕ′(x)dx

∫ 1
0

h1(1− t)h2(t)dt

+f

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))
g

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))∫ 1
0

h1(1− t)h2(1− t)dt.

In the next step we use the right-hand side of (3.4) to estimate
∫b
a
f (x)ϕ′(x)dx and∫b

a
g(x)ϕ′(x)dx and definition of MϕA-h-convexity to estimate f

(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))
and

g
(
ϕ−1

(
ϕ(a)+ϕ(b)

2

))
. After a short calculation we get the required inequality (3.8). �

Remark 3.2. Particular cases of the above results are already known. If ϕ(x) = x, then (3.5) and

(3.6) for AA-h-convex functions are given in [11]. The above results for MϕA-convex functions, i.e.

with h(t) = t are given in [14]. If ϕ(x) = xp, p 6= 0, then (3.5) -(3.8) for MpA-h-convex functions
are given in [5].



Int. J. Anal. Appl. (2022), 20:36 11

The case p =0 is not considered in [5], so, in the following corollary we give this, complementary

result.

Corollary 3.2. Let hi, i = 1,2 be non-negative functions defined on the interval Ji, (0,1) ⊆ Ji, and
let f ,g : [a,b]→ [0,∞), [a,b]⊂ (0,∞).

If f is GA-h1-convex and g is GA-h2-convex, then

(i)

1

logb/a

∫ b
a

f (x)g(x)
dx

x
≤ M(a,b)

∫ 1
0

h1(t)h2(t)dt +N(a,b)

∫ 1
0

h1(t)h2(1− t)dt

(ii)

1

2h1(
1
2
)h2(

1
2
)
f (
√
ab)g(

√
ab)−

1

logb/a

∫ b
a

f (x)g(x)
dx

x

≤ M(a,b)
∫ 1
0

h1(t)h2(1− t)dt +N(a,b)
∫ 1
0

h1(t)h2(t)dt,

where h1(
1
2
)h2(

1
2
) 6=0

(iii)

1

2log2b/a

∫ b
a

∫ b
a

∫ 1
0

1

xy
f (xty1−t)g(xty1−t)dt dy dx

≤
1

logb/a

∫ b
a

f (x)g(x)
dx

x

∫ 1
0

h1(t)h2(t)dt

+[M(a,b)+N(a,b)]

∫ 1
0

h1(t)dt

∫ 1
0

h2(t)dt

∫ 1
0

h1(t)h2(1− t)dt.

Proof. Applying the function ϕ(x)= logx in Theorem 3.2, we get results of this corollary. �

The following theorem also contains some estimations for the integral mean of the product of two

functions, but the proofs of these inequalities are based on the following inequality:

if a ≤ b and c ≤ d, then ad +cb ≤ bd +ac. (3.9)

Theorem 3.3. Let the assumptions of Theorem 3.2 be satisfied. Then

(i) ∫ b
a

[
g(a)h2

(
ϕ(b)−ϕ(x)
ϕ(b)−ϕ(a)

)
+g(b)h2

(
ϕ(x)−ϕ(a)
ϕ(b)−ϕ(a)

)]
f (x)ϕ′(x)dx

+

∫ b
a

[
f (a)h1

(
ϕ(b)−ϕ(x)
ϕ(b)−ϕ(a)

)
+ f (b)h1

(
ϕ(x)−ϕ(a)
ϕ(b)−ϕ(a)

)]
g(x)ϕ′(x)dx

≤ (ϕ(b)−ϕ(a))
[
M(a,b)

∫ 1
0

h1(t)h2(t)dt +N(a,b)

∫ 1
0

h1(t)h2(1− t)dt
]

+

∫ b
a

f (x)g(x)ϕ′(x)dx (3.10)



12 Int. J. Anal. Appl. (2022), 20:36

(ii)

1

ϕ(b)−ϕ(a)

∫ b
a

[
h2

(
1

2

)
f

(
Mϕ(a,b;

1

2
,
1

2
)

)
g(x)

+ h1

(
1

2

)
g

(
Mϕ(a,b;

1

2
,
1

2
)

)
f (x)

]
ϕ′(x)dx

≤
h1
(
1
2

)
h2
(
1
2

)
ϕ(b)−ϕ(a)

∫ b
a

f (x)g(x)ϕ′(x)dx

+h1

(
1

2

)
h2

(
1

2

)[
M(a,b)

∫ 1
0

h1(t)h2(1− t)dt +N(a,b)
∫ 1
0

h1(t)h2(t)dt

]
+
1

2
f

(
Mϕ(a,b;

1

2
,
1

2
)

)
g

(
Mϕ(a,b;

1

2
,
1

2
)

)
, (3.11)

where

M(a,b)= f (a)g(a)+ f (b)g(b), N(a,b)= f (a)g(b)+ f (b)g(a).

Proof. (i) Putting in (3.9)

a = f (Mϕ(a,b;t,1− t)) , b = h1(t)f (a)+h1(1− t)f (b)

c = g (Mϕ(a,b;t,1− t)) , d = h2(t)g(a)+h2(1− t)g(b)

and integrating obtained inequality with respect to t, we get∫ 1
0

[g(a)h2(t)+g(b)h2(1− t)] f (Mϕ(a,b;t,1− t)) dt

+

∫ 1
0

[f (a)h1(t)+ f (b)h1(1− t)]g (Mϕ(a,b;t,1− t)) dt

≤ M(a,b)
∫ 1
0

h1(t)h2(t)dt +N(a,b)

∫ 1
0

h1(t)h2(1− t)dt

+

∫ 1
0

f (Mϕ(a,b;t,1− t))g (Mϕ(a,b;t,1− t)) dt.

After substitution u = Mϕ(a,b;t,1−t) in integrals the above inequality collapses to inequality (3.10).
(ii) From MϕA-hi-convexity we get

f

(
Mϕ(a,b;

1

2
,
1

2
)

)
= f

(
Mϕ(u,v;

1

2
,
1

2
)

)
≤ h1

(
1

2

)
f (u)+h1

(
1

2

)
f (v)

and

g

(
Mϕ(a,b;

1

2
,
1

2
)

)
= g

(
Mϕ(u,v;

1

2
,
1

2
)

)
≤ h2

(
1

2

)
g(u)+h2

(
1

2

)
g(v),

where ϕ(u)= (1− t)ϕ(a)+ tϕ(b) and ϕ(v)= tϕ(a)+(1− t)ϕ(b). Putting in (3.9)

a = f

(
Mϕ(a,b;

1

2
,
1

2
)

)
, b = h1

(
1

2

)
[f (u)+ f (v)]

c = g

(
Mϕ(a,b;

1

2
,
1

2
)

)
, d = h2

(
1

2

)
[g(u)+g(v)]



Int. J. Anal. Appl. (2022), 20:36 13

and integrating with respect to t, we get

h2

(
1

2

)
f

(
Mϕ(a,b;

1

2
,
1

2
)

)∫ 1
0

[g(u)+g(v)]dt

+h1

(
1

2

)
g

(
Mϕ(a,b;

1

2
,
1

2
)

)∫ 1
0

[f (u)+ f (v)]dt

≤ h1
(
1

2

)
h2

(
1

2

)[∫ 1
0

f (u)g(u)dt +

∫ 1
0

f (v)g(v)dt

]
+2h1

(
1

2

)
h2

(
1

2

)[
M(a,b)

∫ 1
0

h1(t)h2(1− t)dt +N(a,b)
∫ 1
0

h1(t)h2(t)dt

]
+f

(
Mϕ(a,b;

1

2
,
1

2
)

)
g

(
Mϕ(a,b;

1

2
,
1

2
)

)
.

After substitution ϕ(x)= (1−t)ϕ(a)+tϕ(b) in integrals
∫1
0
f (u)g(u)dt,

∫1
0
f (u)dt and

∫1
0
g(u)dt,

and substitution ϕ(x)= tϕ(a)+(1− t)ϕ(b) in integrals
∫1
0
f (v)g(v)dt,

∫1
0
f (v)dt and

∫1
0
g(v)dt,

we obtain inequality (3.11). �

Remark 3.3. If h(t)= t, results (3.10) and (3.11) are given in [14].

Conflicts of Interest: The author declares that there are no conflicts of interest regarding the publi-

cation of this paper.

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	1. Preliminaries
	2. Properties of MA-h-convex functions and Jensen-type inequalities
	3. Hermite-Hadamard type inequality and related results
	References