




































©2024 Ada Academica https://adac.eeEur. J. Math. Anal. 4 (2024) 19doi: 10.28924/ada/ma.4.19
Convexity Properties in Non-Newtonian Calculus and Their Applications

Asambo Awini Wilbert1,∗ , Mohammed Muniru Iddrisu2 , Benedict Barnes3
1Department of Mathematics, Bongo Senior High School, Box 7, Bongo District, Upper East Region,

Ghana
awiniwilbert@gmail.com

2Department of Mathematics, Faculty of Physical Sciences, University for Development Studies, Tamale,
Ghana

mmuniru@uds.edu.gh
3Department of Mathematics, Faculty of Physical and Computational Sciences, Kwame Nkrumah

University of Science and Technology, Kumasi, Ghana
ewiekwamina@gmail.com

∗Correspondence: awiniwilbert@gmail.com

Abstract. The study presented some results on convexity properties in non-Newtonian calculus. Alsopresented is the Jensen-Steffensen inequality in non-Newtonian calculus and some applications. Theresearch was mainly on positive real numbers.

1. Introduction
Classical calculus was introduced by Newton and Leibnitz which is applied on our present daymathematics [1]. There are different operations with respect to addition and subtraction of numbersunder this calculus. However, Grossman and Karts came out with another calculus known asnon-Newtonian calculus in the 20th century [2]. Non-Newtonian calculus which is also calledmultiplicative calculus is a multiplicative way of generating positive solutions to mathematicalproblems [3]. It is a recent approach used to solve mathematical problems with positive realnumbers.It is evidently clear that addition is replaced by multiplication in non-Newtonian calculus, andsubtraction by division for example, see authors in [4, 5]. This result has been supported by au-thors in [6], when they introduced the multiplicative calculus and its applications, which has beenestablished to be applicable to solving mathematical problems [6]. Non-Newtonian calculus hasbeen extended in many directions; fractional derivative, complex derivative, integral transformations,differential equations and applications for science and engineering.
Received: 30 Jan 2024.
Key words and phrases. Non-Newtonian calculus, properties, convexity, Jensen-Steffesen inequality.1

https://adac.ee
https://doi.org/10.28924/ada/ma.4.19
https://orcid.org/0009-0004-4991-2067
https://orcid.org/0000-0001-7628-8168
https://orcid.org/0000-0002-0580-5655


The authors in [7] stated that, the centre of all analysis in the social science is the derivative.They were expecting another method which may treat realistic growth phenomenon better in oureconomy than the ordinary approach. The change actually came, which confirmed that variations aremore naturally measured in ratios than in differences [7], until 1972 that Grossman and Katz cameout with non-Newtonian calculus (see [2]). In their work, they also made it clearly that measuringgrowth in ratios gives a better variations than measuring it in differences. Non-Newtonian calculusis essential in the development of our scientific world, which enhances production and development.It is applicable in various ways such as finance (used in marketing and determining rates ofreturn), health (used in tumor therapy and chemotherapy in medicine, pathogen counts in treatedwater), thermostatistics, quantum theory, wave phenomenon, pattern recognition in images (eg. inbiomedicine), signal processing, biology-thus the rate at which growth increases or decays.In the Non-Newtonian calculus, ratios are used in measuring change in values whiles in theclassical approach, differences are used in measuring change in values.
2. Preliminaries

In this section, we give an overview of known definitions and theories used in achieving ourresults.
2.1. Non-Newtonian Arithmetic. A system that satisfies the basic assumptions whose domain is asubset of R is called arithmetic. Exactly one arithmetic result is produced by a generator, whichis a one-to-one function with a range of B that is a subset of the domain R [3,8]. The fundamentalarithmetic operations are defined using the generator as follows [3, 4]:Addition, k+̇r = α [α−1(k) + α−1(r)]Subtraction, k−̇r = α [α−1(k)− α−1(r)]Multiplication, k×̇r = α [α−1(k)× α−1(r)]Division, k/̇r = α [α−1(k)/α−1(r)]When we take α-generator as α(k) = ek , α−1(k) = ln(k) and k = R+, then α arithmetic reducesto non-Newtonian arithmetic as follows:Non-Newtonian - addition,

k+̇r = α
[
α−1(k) + α−1(r)

]
= e(ln(k)+ln(r)) = k · r (1)

Non-Newtonian - subtraction,
k−̇r = α

[
α−1(k)− α−1(r)

]
= e(ln(k)−ln(r)) = k/r (2)

Non-Newtonian - multiplication,
k×̇r = α

[
α−1(k)× α−1(r)

]
= e(ln(k)×ln(r)) = k ln(r) (3)2



Non-Newtonian - division,
k/̇r = α

[
α−1(k)/α−1(r)

]
= e(ln(k)/ ln(r)) = k

1
ln(r) (4)

[1, 4, 8, 9].The above non-Newtonian arithmetic are widely accepted in non-Newtonian calculus.When considering n positive real numbers x1, x2, ..., xn then the α-arithmetic mean is given as [10]:
Aα =

∑n
i=1 xi /̇n =

∑n
i=1 α

[
α−1(xi )
n

]
Aα = α

[
α−1(x1)+α−1(x2)+...+α−1(xn)

n

]
Considering α = exp, we have Aexp = [∏n

i=1 xi
] 1
n = (x1 × x2...xn)

1
nAlso considering x1, x2, ..., xn ∈ R+ and Gα to be the α-geometric mean then:

Gα =
[∏n

i=1 xi
] 1
n = α

[∏n
i=1 α

−1(xi)
] 1
n

Gα = α
[
(α−1(x1)× α−1(x2)...α−1(xn))

1
n

].In a similar way, we take α = exponent, then the α-geometric mean can be interpreted as [10]:
Gexp =

[
(ln x1 × ln x2... ln xn)

1
n

]
, (xn > 1).

Definition 2.1. [11] A set C = [a1, b1] ⊆ R is said to be convex if x, y ∈ C, and

qx + (1− q)y ∈ C, (5)
for q ∈ [0, 1].

Definition 2.2. [11] Let φ be defined on a real interval M . The function φ is convex if:

φ(λ1x + λ2y) ≤ λ1φ(x) + λ2φ(y), (6)
φ(λ1x + (1− λ1)y) ≤ λ1φ(x) + (1− λ1)φ(y), (7)

where λ1 + λ2 = 1, ∀ x, y ∈ M and λ1, λ2 ∈ [0, 1].

Definition 2.3. [12] Let φ be convex function and u1, v1 ∈ R , then

φ

(
u1 + v1
2

)
≤
φ(u1) + φ(v1)

2
. (8)

Proposition 2.1. [13] Let x1 ≤ y1, x2 ≤ y2, and φw be a convex function on an interval of real
positive values, then

φw (y2 − y1)
x2 − x1

≤
φw (y2)− φw (y1)

x2 − x1
. (9)

Theorem 2.1 (Jensen-Steffensen inequality). [14] Let φ be a convex function defined on an interval
of the real line and let xi , pi ∈ R, i = 1, ...m. If x1, ..., xm and p1, ..., pm, pm > 0, then

φ(
1

pm

m∑
i=1

pixi) ≤
1

pm

m∑
i=1

piφ(xi). (10)
3



Theorem 2.2. [14] Let φ be a convex function on an interval S = [a1, b1] ⊂ R+, where a1 < b1.
Let x = (x1, x2, ..., xn) and p = (p1, p2, ..., pn), then

φk(a1 + b1 −
1

pn

n∑
i=1

pixi) ≤ φk(a1) + φk(b1)−
1

pn

n∑
i=1

piφk(xi). (11)
Let pn =

∑n
i=1 pi = 1, then

φk(a1 + b1 −
n∑
i=1

pixi) ≤ φk(a1) + φk(b1)−
n∑
i=1

piφk(xi). (12)
Definition 2.4. [15] The definition of a p-convex set for an interval C is

(qup + (1− q)vp)
1
p ∈ C, (13)

for all u, v ∈ C and q ∈ [0, 1] .

Definition 2.5. [12, 15, 16] Let C = [a1, b1] be an interval on real numbers R. An expression
φ : C = [a1, b1] 7→ R is p-convex if

φ(qup + (1− q)vp)
1
p ≤ qφ(u) + (1− q)φ(v), (14)

∀ u, v ∈ C and q ∈ [0, 1].

Definition 2.6. [15] Let φ be p-convex function and u, v ∈ R+, then

φ

(
up + vp

2

) 1
p

≤
φ(u) + φ(v)

2
. (15)

Definition 2.7. [15] Suppose the function φ : C = [x, y ] 7→ R+ is strongly convex and β ≥ 1; then

φ(qa + (1− q)b) ≤ qφ(a) + (1− q)φ(b)− βq(1− q)(b − a)2 (16)
for all a, b ∈ C and q ∈ [0, 1].

Definition 2.8. [15] A function φ : C = [x, y ] 7→ R is strongly p-convex function, if

φ(qap + (1− q)bp)
1
p ≤ qφ(a) + (1− q)φ(b)− βq(1− q)(bp − ap)2, (17)

for all a, b ∈ C and q ∈ [0, 1].

Definition 2.9. [15] In the event that an interval C is a harmonic convex set, then(
uv

qu + (1− q)v

)
∈ C, (18)

for all u, v ∈ C and q ∈ [0, 1].

Definition 2.10. [15] Let the function φ : C = [a1, b1] ⊆ R+ and C = [a1, b1] be on an interval on
set R+ without zero, then

φ

(
uv

qu + (1− q)v

)
≤ (1− q)φ(u) + qφ(v). (19)

4



Definition 2.11. [15] Let C = [a1, b1] represent an interval on the p-harmonic convex set R without
zero. If a function φ : C = [a1, b1] ⊆ R is p-harmonic convex, it does not include zero if

φ

(
upvp

qup + (1− q)vp

) 1
p

≤ (1− q)φ(u) + qφ(v), (20)
for all u, v ∈ C and q ∈ [0, 1].

In 1984, G. Toader defines m-convex function as follows [17]:
Definition 2.12. Let the function φ be a real R+ on [u, v ] and m ∈ [0, 1], then m-convex function
is given as;

φ
[
qx1 +m(1− q)y1

]
≤ qφ(x1) +m(1− q)φ(y1), (21)

for all x1, y1 ∈ [u, v ] and q ∈ [0, 1]. Also, φ is m-concave if −φ is m−convex.

Definition 2.13. Let the function φ be a positive real value on S = [u, v ], then c-convex is repre-
sented by Luenberger (1969) as;

φ
[
(1− q)x1 + qy1

]
≤ c(1− q)φ(x1) + qφ(y1), (22)

where c ∈ [0, 1], for all x1, y1 ∈ S and q ∈ [0, 1].

3. Results and Discussions
In this section, all the results are presented in non-Newtonian form.

Definition 3.1. Let x, y ∈ C and C = [a1, b1] ⊆ R+ be a set. C must be convex if

x ln(t) · y ln(
1
t
) ∈ C, (23)

for t ∈ [1, e].

Lemma 3.1. Let, x1, y1 ∈ R+ and φ1 be a convex function. Then

φ1

[
x
ln(t1)
1 · y

ln( 1
t1
)

1

]
≤ φ1(x1)ln(t1) · φ1(y1)

ln( 1
t1
)
, (24)

for t1 ∈ [1, e]. 5



Proof. Using equation (3), (1) and by convexity, we have
φ1

[
x
ln(t1)
1 · y

ln( 1
t1
)

1

]
=φ1

[
x1×̇t1+̇y1×̇(

1

t1
)

]
≤φ1(x1)×̇t1+̇φ1(y1)×̇(

1

t1
)

≤α
[
α−1φ1(x1)×̇α−1(t1)+̇α−1φ1(y1)×̇α−1(

1

t1
)

]
≤α

[
lnφ1(x1)× ln(t1) + lnφ1(y1)× ln(

1

t1
)

]
≤e

[
lnφ1(x1) ln(t1)+lnφ1(y1) ln(

1
t1
)
]

≤(e lnφ1(x1))ln(t1) · (e lnφ1(y1))ln(
1
t1
)

φ1

[
x
ln(t1)
1 · y

ln( 1
t1
)

1

]
≤φ1(x1)ln(t1) · φ1(y1)

ln( 1
t1
)
,

as required. �

Lemma 3.2. Let a1, b1 ∈ R+ and φ1 be a convex function, we have

φ1

[
(b1)

(a1)

]
≤
φ1(b1)

φ1(a1)
. (25)

Proof. Using equation (2) and by convexity, we have
φ1

[
(b1)

(a1)

]
=φ1

[
(b1)−̇(a1)

]
≤φ1(b1)−̇φ1(a1)

≤α
[
α−1(φ1(b1))−̇α−1(φ1(a1))

]
≤e[lnφ1(b1)−lnφ1(a1)]

≤
e lnφ1(b1)

e lnφ1(a1)

≤
φ1(b1)

φ1(a1)

φ1

[
(b1)

(a1)

]
≤
φ1(b1)

φ1(a1)
,

as required. �

Lemma 3.3. Let φ1 be convex and u1, v1 ∈ R+. Then

φ1

[
u1 · v1
2

]
≤
φ1(u1) · φ1(v1)

2
. (26)

6



Proof. Using equation (1) and by convexity, we have
φ1

[
u1 · v1
2

]
=φ1

[
1

2
(u1+̇v1)

]
≤
1

2

[(
φ1(u1)+̇φ1(v1)

)]
≤
1

2

[
α
(
α−1φ1(u1)+̇α

−1φ1(v1)
)]

≤
1

2

[
e(lnφ1(u1)+lnφ1(v1))

]
≤
1

2

[
e lnφ1(u1) · e lnφ1(v1)

]
≤
1

2

[
φ1(u1) · φ1(v1)

]
φ1

[
u1 · v1
2

]
≤
φ1(u1) · φ1(v1)

2
,

as required. �

Lemma 3.4. Consider an increasing function φ. If φ(vy ) ≥ φ(vx) and vy ≥ vx , then

φ(vx)
ln(vx ) · φ(vy )ln(vy )

φ(vy )ln(vx ) · φ(vx)ln(vy )
≤ 1, (27)

Proof. Using equation (3) and (1), we have
φ(vx)

ln(vx ) · φ(vy )ln(vy ) =φ(vx)×̇vx +̇φ(vy )×̇vy

=vx ×̇φ(vx)+̇vy ×̇φ(vy )

≤vx ×̇φ(vy )+̇vy ×̇φ(vx)

≤α
[
α−1(vx)×̇α−1φ(vy )+̇α−1(vy )×̇α−1φ(vx)

]
≤α

[
ln(vx)× lnφ(vy ) + ln(vy )× lnφ(vx)

]
≤e[ln(vx )×lnφ(vy )+ln(vy )×lnφ(vx )]

≤e ln(vx ) lnφ(vy ) · e ln(vy ) lnφ(vx )

≤
(
e lnφ(vy )

)ln(vx )
·
(
e lnφ(vx )

)ln(vy )
φ(vx)

ln(vx ) · φ(vy )ln(vy ) ≤φ(vy )ln(vx ) · φ(vx)ln(vy ),as required. �

Lemma 3.5. Consider a decreasing function φ. If φ(vx) ≥ φ(vy ) and vx ≥ vy , then

φ(vx)
ln(vx ) · φ(vy )ln(vy )

φ(vy )ln(vx ) · φ(vx)ln(vy )
≥ 1, (28)

or
φ(vx)

ln(vx ) · φ(vy )ln(vy ) ≥ φ(vy )ln(vx ) · φ(vx)ln(vy ). (29)7



The proof is similar to inequality (25), with inequality sign reversed.
Theorem 3.1. Let vi , zi ∈ R+, i = 1, ..., k , and let φ be a convex function defined on a range of the
real line. If Bk > 0, then

φ

( 1
Bk

)ln∏k
i=1(vi )

ln(zi )
 ≤ ( 1

Bk

)ln∏k
i=1 φ(vi )

ln(zi )

. (30)
Proof. Using equation (3) and by convexity, we have

φ

( 1
Bk

)ln∏k
i=1(vi )

ln(zi )
 =φ 1

Bk
×̇

k∏
i=1

(zi)×̇(vi)


≤
1

Bk
×̇

k∏
i=1

(zi)×̇φ(vi)

≤α

α−1( 1
Bk
)×̇α−1

k∏
i=1

(zi)×̇α−1φ(vi)


≤e

(
ln( 1

Bk
) ln
∏k
i=1(zi )×lnφ(vi )

)

≤
(
e
ln( 1

Bk
)
)ln∏k

i=1(zi )×lnφ(vi )

≤
(
1

Bk

)ln∏k
i=1(zi )×lnφ(vi )

φ

( 1
Bk

)ln∏k
i=1(vi )

ln(zi )
 =( 1

Bk

)ln∏k
i=1 φ(vi )

ln(zi )

,

as required. �

Definition 3.2. A set M = [a1, b1] ⊆ R+ is a p-convex set, if

[
[(x1)

p]ln(j) · [(y1)p]ln(
1
j
)
] 1
p ∈ M, (31)

∀ x1, y1 ∈ M and j ∈ [1, e].

Lemma 3.6. Let x, y ∈ M where M ⊆ R+ and j ∈ [1, e]. For a p-convex function φ, we have

φ
[
[(x1)

p]ln(j) · [(y1)p]ln(
1
j
)
] 1
p ≤

[
φ(x1)

]ln(j) · [φ(y1)]ln( 1j ) . (32)8



Proof. Using equation (3), (1) and by p-convexity, we have
φ
[
[(x1)

p]ln(j) · [(y1)p]ln(
1
j
)
] 1
p
=φ

[
(x1)

p×̇j+̇(y1)p×̇(
1

j
)

] 1
p

≤φ(x1)×̇j+̇φ(y1)×̇(
1

j
)

≤
[
α[α−1φ(x1)×̇α−1(j)+̇α−1φ(y1)×̇α−1(

1

j
)]

]
≤
[
α[lnφ(x1)×̇ ln(j)+̇ lnφ(y1)×̇ ln(

1

j
)]

]
≤e

[
lnφ(x1) ln(j)+lnφ(y1) ln(

1
j
)
]

≤e lnφ(x1) ln(j) · e lnφ(y1) ln(
1
j
)

≤
(
e lnφ(x1)

)ln(j)
·
(
e lnφ(y1)

)ln( 1
j
)

≤
[
φ(x1)

]ln(j) · [φ(y1)]ln( 1j )
φ
[
[(x1)

p]ln(j) · [(y1)p]ln(
1
j
)
] 1
p ≤

[
φ(x1)

]ln(j) · [φ(y1)]ln( 1j ) ,as required. �

Remark 1. The inequality (24) is obtained when p = 1.

Lemma 3.7. Let φ be convex and x1, y1 ∈ R+. For a p-convex function, we have

φ

[
(x1)

p · (y1)p

2

] 1
p

≤
φ(x1) · φ(y1)

2
. (33)

Proof. Using equation (1) and by p-convexity:
φ

[
(x1)

p · (y1)p

2

] 1
p

=φ

[
1

2
((x1)

p+̇(y1)
p)

] 1
p

≤
1

2

[
φ(x1)+̇φ(y1)

]
≤
1

2

[
α
(
α−1φ(x1)+̇α

−1φ(y1)
)]

≤
1

2

[
α
(
lnφ(x1) + lnφ(y1)

)]
≤
1

2

[
e(lnφ(x1)+lnφ(y1))

]
≤
1

2

[
e lnφ(x1) · e lnφ(y1)

]
≤
1

2

[
φ(x1) · φ(y1)

]
φ

[
(x1)

p · (y1)p

2

] 1
p

≤
φ(x1) · φ(y1)

2
,

9



as required. �

Remark 2. The inequality (26) is obtained when p = 1.

Definition 3.3. A harmonic convex set of an interval M is described, if

x ln(y)

x ln(j) · y ln(
1
j
)
∈ M, (34)

∀ x, y ∈ M and j ∈ [1, e].

Lemma 3.8. Let the set W ⊆ R+ be a harmonic set. For harmonic convex function φ, we have

φ

 x
ln(y1)
1

x ln(q1) · y
ln( 1

q1
)

1

 ≤ φ(x1)
lnφ(y1)

φ(x1)ln(q1) · φ(y1)
ln( 1

q1
)
, (35)

∀ x1, y1 ∈ W and q1 ∈ [1, e].

Proof. Using equation (3), (1) and by convexity, we have
φ

 x
ln(y1)
1

x
ln(q1)
1 · y

ln( 1
q1
)

1

 =φ [x1×̇y1−̇(q1×̇x1+̇( 1
q1
)×̇y1)

]

≤φ(x1)×̇φ(y1)−̇(q1×̇φ(x1)+̇(
1

q1
)×̇φ(y1))

≤α
[
α−1φ(x1)×̇α−1φ(y1)−̇(α−1(q1)×̇α−1φ(x1)+̇α−1(

1

q1
)×̇α−1φ(y1))

]
≤α

[
lnφ(x1)× lnφ(y1)− (ln(q1)× lnφ(x1) + ln(

1

q1
)× lnφ(y1))

]
≤e

[
lnφ(x1)×lnφ(y1)−(ln(q1)×lnφ(x1)+ln( 1q1 )×lnφ(y1))

]

≤e
[
lnφ(x1)×lnφ(y1)−(lnφ(x1)×ln(q1)+lnφ(y1)×ln( 1q1 ))

]

≤
(e lnφ(x1))lnφ(y1)

(e lnφ(x1))ln(q1) · (e lnφ(y1))ln(
1
q1
)

φ

 x
ln(y1)
1

x
ln(q1)
1 · y

ln( 1
q1
)

1

 ≤ φ(x1)
lnφ(y1)

φx
ln(q1)
1 · φ(y1)

ln( 1
q1
)
,

proved. �

Definition 3.4. Let W be a subset on R+, then W is p-harmonic convex set if [xp]ln(y)
p

[xp]ln(j) · [yp]ln(
1
j
)

 1p ∈ W, (36)
∀ x, y ∈ W and j ∈ [1, e]. 10



Lemma 3.9. Consider the p-harmonic convex set M = [a, b] ⊆ R+. If φ is p-harmonic convex
function, we have

φ

 [(x1)
p]ln(y1)

p

[(x1)p]ln(j1) · [(y1)p]
ln( 1

j1
)

 1p ≤ [φ(x1)]
lnφ(y1)

[φ(x1)]ln(j) · [φ(y1)]
ln( 1

j1
)
, (37)

∀ x1, y1 ∈ M and j1 ∈ [1, e].

Proof. Using equation (3), (2) and by p-convexity, we have
φ

 [(x1)
p]ln(y1)

p

[(x1)p]ln(j1) · [(y1)p]
ln( 1

j1
)

 1p =φ [(x1)p×̇(y1)p−̇(j1×̇(x1)p+̇( 1
j1
)×̇(y1)p)

] 1
p

≤φ(x1)×̇φ(y1)−̇(j1×̇φ(x1)+̇(
1

j1
)×̇φ(y1))

≤α
[
lnφ(x1)× lnφ(y1)− (ln(j1)× lnφ(x1) + ln(

1

j1
)× lnφ(y1)

]
≤e

[
lnφ(x1)×lnφ(y1)−(ln(j1)×lnφ(x1)+ln( 1j1 )×lnφ(y1)

]

≤e
[
lnφ(x1)×lnφ(y1)−(lnφ(x1)×ln(j1)+lnφ(y1)×ln( 1j1 )

]

≤
e[lnφ(x1) lnφ(y1)]

e[lnφ(x1) ln(j1)] · e[lnφ(y1) ln(
1
j1
)]

≤
(φ(x1))

lnφ(y1)

(φ(x1))ln(j1) · (φ(y1))
ln( 1

j1
)

φ

 [(x1)
p]ln(y1)

p

[(x1)p]ln(j1) · [(y1)p]
ln( 1

j1
)

 1p ≤ [φ(x1)]
lnφ(y1)

[φ(x1)]ln(j1) · [φ(y1)]
ln( 1

j1
)
,

as required. �

Remark 3. When p = 1, the inequality (35) is obtained.

Definition 3.5. M is referred to as p-Jensen-Steffensen’s set if, it is a subset of R+, assuming that( 1
Bn

)ln∏n
i=1((xi )

p)ln(zi )
 1p ∈ M. (38)

Lemma 3.10. Let M ⊆ R+ be p-Jensen-Steffensen set. For p-Jensen-Steffensen’s inequality, we
have

φ

( 1
Bn

)ln∏n
r=1((xi )

p)ln(zi )
 1p ≤ ( 1

Bn

)ln∏n
r=1 φ(xi )

ln(zi )

. (39)
11



Proof. Using equation (3) and by p-convexity, we have

φ

( 1
Bn

)ln∏n
r=1((xi )

p)ln(zi )
 1p =φ 1

Bn
×̇

n∏
r=1

(zi)×̇(xi)p
 1p

≤
1

Bn
×̇

n∏
r=1

(zi)×̇φ(xi)

≤α

α−1( 1
Bn
)×̇α−1

n∏
r=1

(zi)×̇α−1φ(xi)


≤e

(
ln( 1

Bn
) ln
∏n
r=1(zi )×lnφ(xi )

)

≤
(
e ln(

1
Bn
)
)ln∏n

r=1(zi )×lnφ(xi )

≤
(
1

Bn

)ln∏n
r=1(zi )×lnφ(xi )

φ

( 1
Bn

)ln∏n
r=1((xi ))

ln(zi )
 1p ≤( 1

Bn

)ln∏n
r=1 φ(xi )

ln(zi )

,

as required. �

Definition 3.6. Let the set W be a subset on R+, then W is strongly convex set if

[[
µln(j)

]ln( 1
j
)
]ln( y

x
)2

∈ W, (40)
where µ ≥ 1, ∀ x, y ∈ W and j ∈ [1, e].

Lemma 3.11. Consider a convex set W ⊆ R+. For a strongly convex function φ, we have

φ
[
x ln(q) · y ln(

1
q
)
]
≤
φ(x)ln(q) · φ(y)ln(

1
q
)[[

µln(q)
]ln( 1

q
)
]ln( y

x
)2
, (41)

where µ ≥ 1, ∀ x, y ∈ W and q ∈ [1, e]. 12



Proof. Using equation (3), (1) and by convexity, we have
φ
[
x ln(q) · y ln(

1
q
)
]
=φ

[
x×̇q+̇y×̇(

1

q
)

]
≤q×̇φ(x)+̇(

1

q
)×̇φ(y)−̇µ×̇j×̇(

1

q
)×̇(

y

x
)2

≤α
[
α−1q×̇α−1φ(x)+̇α−1(

1

q
)×̇α−1φ(y)−̇α−1(µ)×̇α−1(q)×̇α−1(

1

q
)×̇(

y

x
)2
]

≤
e ln(q) lnφ(x) · e ln(

1
q
) lnφ(y)

e ln(µ) ln(q) ln(
1
q
) ln( y

x
)2)

≤

(
e lnφ(x)

)ln(q)
·
(
e lnφ(y)

)ln( 1
q
)

(
e ln(µ)

)ln(q) ln( 1
q
) ln( y

x
)2

φ
[
x ln(q) · y ln(

1
q
)
]
≤
φ(x)ln(q) · φ(y)ln(

1
q
)[[

µln(q)
]ln( 1

q
)
]ln( y

x
)2
,

proved. �

Lemma 3.12. Consider the convex set W = [a, b] ⊆ R+. If φ is highly p-convex function, we have

φ
[
(xp)ln(j) · (yp)ln(

1
j
)
] 1
p ≤

φ(x)ln(j) · φ(y)ln(
1
j
)[[

µln(j)
]ln( 1

j
)
]ln( y

x
)2
, (42)

where µ ≥ 1, ∀ x, y ∈ W and j ∈ [1, e].
Remark 4. When p = 1, the strongly p-convex function returns to strongly convex function. Thus
the inequality (41) is obtained.

Proof. Using equation (3), (1) and by p-convexity, we have
φ
[
(xp)ln(j) · (yp)ln(

1
j
)
] 1
p ≤j×̇φ(x)+̇(

1

j
)×̇φ(y)−̇µ×̇j×̇(

1

j
)×̇(

y

x
)2

≤α
[
α−1j×̇α−1φ(x)+̇α−1(

1

j
)×̇α−1φ(y)−̇α−1(µ)×̇α−1(j)×̇α−1(

1

j
)×̇(

y

x
)2
]

≤e
(
ln(j) lnφ(x)+ln( 1

j
) lnφ(y)−ln(µ) ln(j) ln( 1

j
) ln( y

x
)2
)

≤
e ln(j) lnφ(x) · e ln(

1
j
) lnφ(y)

e ln(µ) ln(j) ln(
1
j
) ln( y

x
)2)

≤

(
e lnφ(x)

)ln(j)
·
(
e lnφ(y)

)ln( 1
j
)

(
e ln(µ)

)ln(j) ln( 1
j
) ln( y

x
)2

13



φ
[
(xp)ln(j) · (yp)ln(

1
j
)
] 1
p ≤

φ(x)ln(j) · φ(y)ln(
1
j
)[[

µln(j)
]ln( 1

j
)
]ln( y

x
)2
,

proved. �

Definition 3.7. Let the set W be a subset on R+, then W is m-convex set if

x ln(t) ·
[
(y)ln(m)

]ln( 1
t
)
∈ W, (43)

where m ∈ [1, e], ∀ x, y ∈ W and j ∈ [1, e].

Lemma 3.13. Let the function φ1 be m-convex and m1 ∈ [1, e], then

φ1

[
x
ln(t1)
1 ·

[
y
ln(m)
1

]ln( 1
t1
)
]
≤ φ1(x1)ln(t1) ·

[
φ1(y1)

ln(m)
]ln( 1

t1
)
, (44)

for all x1, y1 ∈ R+ and t ∈ [1, e].
Proof. Using equation (3), (1) and by convexity, we have
φ1

[
x
ln(t1)
1 ·

[
(y1)

ln(m1)
]ln( 1

t1
)
]
=φ1

[
x1×̇t+̇m1×̇(y1)×̇(

1

t1
)

]
≤φ1(x1)×̇t1+̇m1×̇φ1(y1)×̇(

1

t1
)

≤
[
α[α−1φ1(x1)×̇α−1(t1)+̇α−1(m1)×̇α−1φ1(y1)×̇α−1(

1

t1
)]

]
≤
[
α[lnφ1(x1)× ln(t1) + ln(m1)×̇ lnφ1(y1)× ln(

1

t1
)]

]
≤
[
e
[lnφ1(x1)×ln(t1)+ln(m1)×lnφ1(y1)×ln( 1t1 )]

]
≤
[
e
[lnφ1(x1) ln(t1)+lnm1 lnφ1(y1) ln(

1
t1
)]
]

≤(e lnφ1(x1))ln(t1) · ((e lnφ1(y1))ln(m1))ln(
1
t1
)

φ1

[
x
ln(t1)
1 ·

[
(y1)

ln(m1)
]ln( 1

t
)
]
≤φ1(x1)ln(t1) ·

[
φ1(y1)

ln(m1)
]ln( 1

t1
)
,

as required. �

Definition 3.8. Let W ⊆ R+, then W is said to be c-convex set if

x
ln( 1

t
)

1 · y ln(t)1 ∈ W, (45)
where c ∈ [1, e], ∀ x1, y1 ∈ W and t ∈ [1, e]. 14



Lemma 3.14. Let the function φ1 be c-convex on W = [a, a1] and c ∈ [1, e], then

φ1

[
(x
ln( 1

t1
)

1 · y ln(t1)1

]
≤
[
φ1(x1)

ln(c)
]ln( 1

t1
)
· φ1(y1)ln(t1), (46)

∀ x1, y1 ∈ W and t1 ∈ [1, e].

Proof. Using equation (3), (1) and by convexity, we have
φ1

[
x
ln( 1

t1
)

1 · (y1)ln(t1)
]
=φ1

[
x1×̇(

1

t1
)+̇(y1)×̇t1

]
≤c×̇φ1(x1)×̇(

1

t1
)+̇φ1(y1)×̇t1

≤
[
α[α−1(c)×̇α−1φ1(x1)×̇α−1(

1

t1
)+̇α−1φ1(y1)×̇α−1(t1)]

]
≤
[
α[ln(c)× lnφ1(x1)× ln(

1

t1
) + lnφ1(y1)× ln(t1)]

]
≤
[
e
[lnφ1(x1)×ln(c)×ln( 1t1 )+lnφ1(y1)×ln(t1)]

]
≤
[
e
[lnφ1(x1) ln(c) ln(

1
t1
)+lnφ1(y1) ln(t1)]

]
≤
[
(e lnφ1(x1))ln(c)

]ln( 1
t1
)
· (e lnφ1(y1))ln(t1)

φ1

[
x
ln( 1

t1
)

1 · (y1)ln(t1)
]
≤
[
φ1(x1)

ln(c)
]ln( 1

t1
)
· φ1(y1)ln(t1),as required. �

4. Conclusion
In this paper, some classes of convex functions have been identified and presented. The pa-per established some convexity properties and inequalities in non-Newtonian calculus and theirapplications.

References
[1] E. Unluyol, S. Salas, İ. İscan, Convex functions and some inequalities in terms of the non-newtonian calculus, in:AIP Conference Proceedings, Vol. 1833, AIP Publishing LLC, 2017, p. 020043.[2] A. E. Bashirov, R. Mustafa, On complex multiplicative differentiation, TWMS Journal of applied and engineeringmathematics 1 (1) (2011) 75–85.[3] K. Boruah, B. Hazarika, G-calculus, TWMS Journal of Applied and Engineering Mathematics 8 (1) (2018) 94–105.[4] D. F. Torres, On a non-newtonian calculus of variations, Axioms 10 (3) (2021) 171.[5] M. Czachor, Non-newtonian mathematics instead of non-newtonian physics: Dark matter and dark energy from amismatch of arithmetics, Foundations of Science 26 (2021) 75–95.[6] A. E. Bashirov, E. M. Kurpınar, A. Özyapıcı, Multiplicative calculus and its applications, Journal of mathematicalanalysis and applications 337 (1) (2008) 36–48.[7] D. Filip, C. Piatecki, An overview on the non-newtonian calculus and its potential applications to economics.15



[8] M. Grossman, Bigeometric calculus: a system with a scale-free derivative, Archimedes Foundation, 1983.[9] M. Grossman, R. Katz, Non-Newtonian Calculus: A Self-contained, Elementary Exposition of the Authors’ Investi-gations..., Non-Newtonian Calculus, 1972.[10] U. Kadak, Y. Gürefe, A generalization on weighted means and convex functions with respect to the non-newtoniancalculus, International Journal of Analysis 2016.[11] M. A. Noor, K. I. Noor, S. Iftikhar, Hermite-hadamard inequalities for harmonic nonconvex functions, MAGNT Res.Rep 4 (2016) 24–40.[12] S. S. Dragomir, n-points inequalities of hermite-hadamard type for h-convex functions on linear spaces, ArmenianJournal of Mathematics 8 (1) (2016) 38–57.[13] I. Franjić, S. Khalid, J. Pečarić, On the refinements of the jensen-steffensen inequality, Journal of inequalities andApplications 2011 (1) (2011) 1–11.[14] M. Bakula, M. Matić, J. Pečarić, Generalizations of the jensen-steffensen and related inequalities, Open Mathematics7 (4) (2009) 787–803.[15] H. Li, M. S. Saleem, I. Ahmed, K. N. Aslam, Hermite–hadamard and fejér-type inequalities for strongly reciprocally(p, h)-convex functions of higher order, Journal of Inequalities and Applications 2023 (1) (2023) 1–20.[16] I. Iscan, Ostrowski type inequalities for p-convex functions, New Trends in Mathematical Sciences 4 (3) (2016)140–150.[17] J. N. Valdés, F. Rabossi, A. D. Samaniego, Convex functions: Ariadne’s thread or charlotte’s spiderweb, AdvancedMathematical Models & Applications 5 (2) (2020) 176–191.

16


	1. Introduction
	2. Preliminaries
	2.1. Non-Newtonian Arithmetic

	3. Results and Discussions
	4. Conclusion
	References

