Startseite Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
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Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions

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Demonstratio Mathematica
Aus der Zeitschrift Demonstratio Mathematica Band 58 Heft 1

Abstract

In this study, we establish new generalizations and results for Simpson, midpoint, and trapezoid-type integral inequalities within the framework of multiplicative calculus. We begin by proving a new identity for multiplicatively differentiable functions. Using this identity, we then obtain a new Simpson-type inequality for multiplicatively s -convex functions. Additionally, we derive novel integral inequalities related to the right and left sides of the Hermite-Hadamard inequality for multiplicatively s -convex functions. We also demonstrate that some of the results obtained here improve upon known results, while others are generalizations. Finally, we present some applications to special means.

Keywords: multiplicative calculus; multiplicatively s-convex function; Simpson inequality; midpoint-type inequality; trapezoid-type inequality
MSC 2010: 26D07; 26D10; 26D15; 26A51

1 Introduction

Let I be an interval of real numbers. A function f : I → R is said to be convex, if

(1) f ( t ϰ + ( 1 − t ) y ) ≤ t f ( ϰ ) + ( 1 − t ) f ( y )

holds for all ϰ , y ∈ I and all t ∈ [ 0 , 1 ] .

Convexity of functions is a powerful approach extensively employed to address various challenges in both pure and applied research. The application of convexity has led to the discovery of numerous extensions and generalizations of integral inequalities, along with their valuable applications [1–19]. Several studies have demonstrated that many of the results obtained concerning these inequalities are direct outcomes of the applications of convex functions. The Hermite-Hadamard inequality, which is a highly important mathematical inequality associated with convex functions, finds widespread application in numerous other fields of mathematics.

The theory of convexity is a fundamental aspect in various domains, such as mathematical finance, economics, engineering, management sciences, and optimization theory. Several studies have demonstrated that many of the results obtained concerning these inequalities are direct outcomes of the applications of convex functions. The Hermite-Hadamard inequality stands as one of the most renowned inequalities related to the integral mean of a convex function. This double inequality is expressed as follows (see, [20–22]).

Theorem 1

Let f : I ⊆ R → R be a convex function and a , b ∈ I with a < b . Then,

(2) f a + b 2 ≤ 1 b − a ∫ a b ( f ( ϰ ) ) d ϰ ≤ f ( a ) + f ( b ) 2 .

Both inequalities hold in the reversed direction if ϒ is concave.

The Hermite-Hadamard inequality can be viewed as an enhancement of the concept of convexity. In recent times, several generalizations and extensions have been explored for classical convexity. Additionally, numerous investigations have been dedicated to establishing new bounds for both the left and right-hand sides of inequality (2). For some illustrative instances, please refer the monographs [23–33].

The following inequality, identified as Simpson’s inequality, ranks among the most widely recognized outcomes in the literature.

Theorem 2

Let f : [ a , b ] → R be a four times continuously differentiable mapping on ( a , b ) and ∥ f ( 4 ) ∥ ∞ = sup ϰ ∈ ( a , b ) ∣ f ( 4 ) ( ϰ ) ∣ < ∞ . Then, the following inequality holds:

(3) 1 3 f ( a ) + f ( b ) 2 + 2 f a + b 2 − 1 b − a ∫ a b f ( ϰ ) d ϰ ≤ 1 2,880 ∥ f ( 4 ) ∥ ∞ ( b − a ) 4 .

For some extensions and generalizations of the Simpson’s-type inequalities using novel and innovative methods, refer [34–47] and the corresponding references cited therein.

Dragomir et al. [34] gave the following Simpson’s inequality for which the remainder is expressed in terms of derivatives lower than the fourth.

Theorem 3

Let f : [ a , b ] → R be a differentiable mapping whose derivative is continuous on ( a , b ) and f ′ ∈ L [ a , b ] . Then, the following inequality holds:

1 3 f ( a ) + f ( b ) 2 + 2 f a + b 2 − 1 b − a ∫ a b f ( ϰ ) d ϰ ≤ b − a 3 ∥ f ′ ∥ 1

where ∥ f ′ ∥ 1 = ∫ a b ∣ f ′ ( ϰ ) ∣ d ϰ .

In [34], the bound of inequality (3) for L -Lipschitzian mapping was given by 5 36 L ( b − a ) .

In the investigation of inequalities, the multiplicatively convex function is of paramount importance, and in [22], it is defined as follows.

Definition 4

A function f : I → ( 0 , ∞ ) is said to be multiplicatively convex, if

f ( ( 1 − t ) ϰ + t y ) ≤ [ f ( ϰ ) ] 1 − t [ f ( y ) ] t

for all ϰ , y ∈ I and t ∈ [ 0 , 1 ] .

It can be easily seen that

f ( t ϰ + ( 1 − t ) y ) ≤ [ f ( ϰ ) ] 1 − t [ f ( y ) ] t ≤ t f ( ϰ ) + ( 1 − t ) f ( y ) ,

which implies that every multiplicatively convex function is a convex function; however, the opposite is not true.

In [48], Hermite-Hadamard inequality for multiplicatively convex functions is given as follows.

Theorem 5

Let f be a positive and multiplicatively convex function on [ a , b ] . Then,

f a + b 2 ≤ ∫ a b ( f ( ϰ ) ) d ϰ 1 b − a ≤ G ( f ( a ) , f ( b ) ) ,

where G ( ⋅ , ⋅ ) is the geometric mean.

Multiplicatively convex functions have been generalized into various types of multiplicatively convex functions by different mathematicians until today [49–57]. One of these generalizations was obtained by Xi and Qi [33].

Definition 6

A function f : I → ( 0 , ∞ ) is said to be multiplicatively s -convex, where s ∈ ( 0 , 1 ] , if

(4) f ( ( 1 − t ) ϰ + t y ) ≤ [ f ( ϰ ) ] ( 1 − t ) s [ f ( y ) ] t s

for all ϰ , y ∈ I and t ∈ [ 0 , 1 ] .

It is obvious that for s = 1 , inequality (4) reduces to inequality (1).

2 Multiplicative calculus

During the seventeenth century, Isaac Newton and Gottfried Wilhelm Leibniz separately unveiled the principles of differential and integral calculus. These groundbreaking discoveries have since played a pivotal role in the realm of analysis and calculus. The fundamental operations involved, differentiation and integration, permit the manipulation of numbers at an infinitesimal level.

In a significant departure from convention, a novel form of calculus emerged between 1967 and 1970 through the pioneering efforts of Grossman and Katz. This innovative framework reimagined the traditional notions of addition and subtraction by introducing division and multiplication as the central operations. Referred to as “multiplicative calculus,” this new approach is tailored exclusively to positive functions. Due to its narrower scope of application, multiplicative calculus has not garnered the same level of recognition as the calculus conceived by Leibniz and Newton. Nonetheless, it has revealed intriguing outcomes across various domains.

The foundations of multiplicative calculus were initially laid out by Bashirov et al. [58] in the 1970s, as they revisited and modified the classical calculus proposed by Newton and Leibniz. While its domain of applicability remains constrained to positive functions, multiplicative calculus has yielded remarkable results, evidenced by its diverse applications.

For instance, Bashirov et al. [59] introduced a fundamental theorem that underpins multiplicative calculus. Furthermore, Bashirov et al. [59] saw Bashirov and Riza pioneering the concept of complex multiplicative calculus. The realm of stochastic multiplicative calculus was explored by Bashirov and Ríza [60], where various properties were examined.

For a comprehensive exploration of the applications and dimensions within this discipline, one can refer to [61–65] and references therein.

2.1 Multiplicative derivatives and integrals

Now, let us recall certain definitions, properties, and concepts related to differentiation and multiplicative integration.

Definition 7

[59] Let f : R → R +  be a positive function. The multiplicative derivative of the function f noted by f * is defined as follows:

d * f d t = f * ( t ) = lim h → 0 f ( t + h ) f ( t ) 1 h .

Remark 8

If f has positive values and is differentiable at t , then f * exists and the relation between f * and ordinary derivative f ′ is as follows:

f * ( t ) = e ( ln f ( t ) ) ′ = e f ′ ( t ) f ( t ) .

The multiplicative derivative admits the following properties.

Theorem 9

[59] Let f and g be multiplicatively differentiable functions, and c is the arbitrary constant. Then, functions c f , f g , f + g , f ⁄ g , and f g are * differentiable and

  1. ( c f ) * ( t ) = f * ( t ) ,

  2. ( f g ) * ( t ) = f * ( t ) g * ( t ) ,

  3. ( f + g ) * ( t ) = f * ( t ) f ( t ) f ( t ) + g ( t ) g * ( t ) g ( t ) f ( t ) + g ( t ) ,

  4. f g * ( t ) = f * ( t ) g * ( t ) ,

  5. ( f g ) * ( t ) = f * ( t ) g ( t ) f ( t ) g ′ ( t ) .

It is worth noting that the concept of the multiplicative integral, denoted as * integral, takes the form ∫ a b ( f ( ϰ ) ) d ϰ , diverging from the classical Riemann integral represented as ∫ a b ( f ( ϰ ) ) d ϰ . This distinction arises from the utilization of different summation methods. Specifically, the classical Riemann integral employs term summation, while the * integral relies on products of raised terms within the interval [ a , b ] .

The relationship connecting the Riemann integral to the * integral can be expressed as follows.

Proposition 10

[59] If f is Riemann integrable on [ a , b ] , then f is * integrable on [ a , b ] and

∫ a b ( f ( ϰ ) ) d ϰ = e ∫ a b ln ( f ( ϰ ) ) d ϰ .

* integral yields the following results and properties:

Theorem 11

[59] Let f be positive and Riemann integrable on [ a , b ] , then f is * integrable on [ a , b ] and

  1. ∫ a b ( ( f ( ϰ ) ) p ) d ϰ = ∫ a b ( f ( ϰ ) ) d ϰ p ,

  2. ∫ a b ( f ( ϰ ) g ( ϰ ) ) d ϰ = ∫ a b ( f ( ϰ ) ) d ϰ ∫ a b ( g ( ϰ ) ) d ϰ ,

  3. ∫ a b f ( ϰ ) g ( ϰ ) d ϰ = ∫ a b ( f ( ϰ ) ) d ϰ ∫ a b ( g ( ϰ ) ) d ϰ ,

  4. ∫ a b ( f ( ϰ ) ) d ϰ = ∫ a c ( f ( ϰ ) ) d ϰ ∫ c b ( f ( ϰ ) ) d ϰ , a < c < b ,

  5. ∫ a a ( f ( ϰ ) ) d ϰ = 1 and ∫ a b ( f ( ϰ ) ) d ϰ = ∫ b a ( f ( ϰ ) ) d ϰ − 1 .

* integration by parts is given as follows.

Theorem 12

[59] Let f : [ a , b ] → R be multiplicatively differentiable, let g : [ a , b ] → R be differentiable so the function f g is * integrable, and

∫ a b ( f * ( ϰ ) g ( ϰ ) ) d ϰ = f ( b ) g ( b ) f ( a ) g ( a ) × 1 ∫ a b ( f ( ϰ ) g ′ ( ϰ ) ) d ϰ .

In order to prove our results, we need the following lemmas.

Lemma 13

[50] Let f : [ a , b ] → R + be a * differentiable mapping on [ a , b ] with a < b . If f ∗ is * integrable on [ a , b ] , then

f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b = ∫ 0 1 f ∗ ( 1 − t ) a + t a + b 2 t d t b − a 4 ∫ 0 1 f ∗ ( 1 − t ) a + b 2 + t b t − 1 d t b − a 4 .

Lemma 14

[54] Let f : [ a , b ] → R + be a * differentiable mapping on [ a , b ] with a < b . If f ∗ is * integrable on [ a , b ] , then

G ( f ( a ) , f ( b ) ) ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b = ∫ 0 1 f ∗ ( ( 1 − t ) a + t b ) 1 2 − t d t b − a ,

where G is the geometric mean.

3 Main results

In this section, we establish some new Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s -convex functions. First we give the following lemma.

Lemma 15

Let f : [ a , b ] → R + be a * differentiable mapping on [ a , b ] with a < b . If f ∗ is * integrable on [ a , b ] , then

( f ( a ) ) 1 6 f a + b 2 2 3 ( f ( b ) ) 1 6 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b = ∫ 0 1 f ∗ 1 + t 2 b + 1 − t 2 a t 2 − 1 3 d t b − a 2 ∫ 0 1 f ∗ 1 + t 2 a + 1 − t 2 b 1 3 − t 2 d t b − a 2 .

Proof

Let

I 1 = ∫ 0 1 f ∗ 1 + t 2 b + 1 − t 2 a t 2 − 1 3 d t b − a 2

and

I 2 = ∫ 0 1 f ∗ 1 + t 2 a + 1 − t 2 b 1 3 − t 2 d t b − a 2 .

Using the integration by parts for * integrals, we have

I 1 = ∫ 0 1 f ∗ 1 + t 2 b + 1 − t 2 a t 2 − 1 3 d t b − a 2

= ∫ 0 1 f ∗ 1 + t 2 b + 1 − t 2 a b − a 2 t 2 − 1 3 d t = ( f ( b ) ) 1 6 f a + b 2 − 1 3 . 1 ∫ 0 1 f 1 + t 2 b + 1 − t 2 a 1 2 d t = ( f ( b ) ) 1 6 f a + b 2 1 3 ∫ 0 1 f 1 + t 2 b + 1 − t 2 a d t − 1 2 = ( f ( b ) ) 1 6 f a + b 2 1 3 ∫ a + b 2 b ( f ( ϰ ) ) d ϰ 1 a − b ,

I 2 = ∫ 0 1 f ∗ 1 + t 2 a + 1 − t 2 b 1 3 − t 2 d t b − a 2 = ∫ 0 1 f ∗ 1 + t 2 a + 1 − t 2 b a − b 2 t 2 − 1 3 d t = ( f ( a ) ) 1 6 f a + b 2 − 1 3 . 1 ∫ 0 1 f 1 + t 2 a + 1 − t 2 b 1 2 d t = ( f ( a ) ) 1 6 f a + b 2 1 3 ∫ 0 1 f 1 + t 2 a + 1 − t 2 b d t − 1 2 = ( f ( a ) ) 1 6 f a + b 2 1 3 ∫ a a + b 2 ( f ( ϰ ) ) d ϰ 1 a − b .

Multiplying above equalities, we have

I 1 × I 2 = ( f ( b ) ) 1 6 f a + b 2 1 3 ∫ a + b 2 b ( f ( ϰ ) ) d ϰ 1 a − b ( f ( a ) ) 1 6 f a + b 2 1 3 ∫ a a + b 2 ( f ( ϰ ) ) d ϰ 1 a − b = ( f ( a ) ) 1 6 f a + b 2 2 3 ( f ( b ) ) 1 6 ∫ a a + b 2 ( f ( ϰ ) ) d ϰ ∫ a + b 2 b ( f ( ϰ ) ) d ϰ 1 a − b = ( f ( a ) ) f a + b 2 4 ( f ( b ) ) 1 6 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b .

So, the proof is completed.□

Now, we are in a position to derive new integral inequalities for multiplicatively s -convex functions related to multiplicative integrals.

Theorem 16

Let f : [ a , b ] → R + be a * differentiable mapping on [ a , b ] with a < b . If f ∗ is multiplicatively s-convex on [ a , b ] , then the following Simpson-type inequality holds:

(5) ( f ( a ) ) f a + b 2 4 ( f ( b ) ) 1 6 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( ( f ∗ ( a ) ) ( f ∗ ( b ) ) ) ( s − 4 ) 6 s + 1 + 2 × 5 s + 2 − 2 × 3 s + 2 + 2 6 s + 2 ( s + 1 ) ( s + 2 ) ( b − a ) .

Proof

From Lemma 15 and properties of * integral, we obtain

( f ( a ) ) f a + b 2 4 ( f ( b ) ) 1 6 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ exp b − a 2 ∫ 0 1 ln f ∗ 1 + t 2 b + 1 − t 2 a t 2 − 1 3 d t × exp b − a 2 ∫ 0 1 ln f ∗ 1 + t 2 a + 1 − t 2 b 1 3 − t 2 d t = exp b − a 2 ∫ 0 1 t 2 − 1 3 ln f ∗ 1 + t 2 b + 1 − t 2 a d t × exp b − a 2 ∫ 0 1 1 3 − t 2 ln f ∗ 1 + t 2 a + 1 − t 2 b d t .

By using the multiplicative s -convexity of f ∗ , we have

( f ( a ) ) f a + b 2 4 ( f ( b ) ) 1 6 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ exp b − a 2 ∫ 0 1 t 2 − 1 3 ln ( f ∗ ( b ) ) 1 + t 2 s ( f ∗ ( a ) ) 1 − t 2 s d t × exp b − a 2 ∫ 0 1 1 3 − t 2 ln ( f ∗ ( a ) ) 1 + t 2 s ( f ∗ ( b ) ) 1 − t 2 s d t = exp b − a 2 ∫ 0 1 t 2 − 1 3 1 + t 2 s ln ( f ∗ ( b ) ) + 1 − t 2 s ln ( f ∗ ( a ) ) d t × exp b − a 2 ∫ 0 1 1 3 − t 2 1 + t 2 s ln ( f ∗ ( a ) ) + 1 − t 2 s ln ( f ∗ ( b ) ) d t = exp b − a 2 s + 1 ∫ 0 1 t 2 − 1 3 [ ( 1 + t ) s + ( 1 − t ) s ] [ ln ( f ∗ ( a ) ) + ln ( f ∗ ( b ) ) ] d t .

It is easy to observe that

∫ 0 1 t 2 − 1 3 [ ( 1 + t ) s + ( 1 − t ) s ] d t = ∫ 0 2 3 1 3 − t 2 [ ( 1 + t ) s + ( 1 − t ) s ] d t + ∫ 2 3 1 t 2 − 1 3 [ ( 1 + t ) s + ( 1 − t ) s ] d t = J 1 + J 2 .

J 1 = ∫ 0 2 3 1 3 − t 2 [ ( 1 + t ) s + ( 1 − t ) s ] d t = ∫ 0 2 3 1 3 − t 2 ( 1 + t ) s d t + ∫ 0 2 3 1 3 − t 2 ( 1 − t ) s d t = 5 ( 1 + t ) s + 1 6 ( s + 1 ) − ( 1 + t ) s + 2 2 ( s + 2 ) 0 2 3 + 5 ( 1 + t ) s + 1 6 ( s + 1 ) − ( 1 + t ) s + 2 2 ( s + 2 ) 0 2 3 = 5 s + 2 − 2 × 3 s + 2 + 1 2 × 3 s + 2 ( s + 1 ) ( s + 2 ) ,

J 2 = ∫ 2 3 1 t 2 − 1 3 [ ( 1 + t ) s + ( 1 − t ) s ] d t = ∫ 2 3 1 t 2 − 1 3 ( 1 + t ) s d t + ∫ 2 3 1 t 2 − 1 3 ( 1 − t ) s d t = ( 1 + t ) s + 2 2 ( s + 2 ) − 5 ( 1 + t ) s + 1 6 ( s + 1 ) 2 3 1 + − ( 1 + t ) s + 1 6 ( s + 1 ) + ( 1 − t ) s + 2 2 ( s + 2 ) 2 3 1 = ( s − 4 ) 6 s + 1 + 5 s + 2 + 1 2 × 3 s + 2 ( s + 1 ) ( s + 2 ) .

So, the proof is completed.□

Corollary 17

If we take s = 1 in Theorem 16, then f ∗ is a multiplicatively convex function on [ a , b ] and inequality (5) reduces to the following inequality:

( f ( a ) ) f a + b 2 4 ( f ( b ) ) 1 6 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( ( f ∗ ( a ) ) ( f ∗ ( b ) ) ) 5 ( b − a ) 72 .

Corollary 18

In Corollary 17, if f ( a ) = f a + b 2 = f ( b ) , then we have

(6) f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( ( f ∗ ( a ) ) ( f ∗ ( b ) ) ) 5 ( b − a ) 72 .

Corollary 19

In Corollary 18, if f ∗ ≤ K , then we have

f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ K 5 ( b − a ) 36 .

Remark 20

We note that the obtained inequality (6) is better than the inequality in Theorem 3.3 in [54].

Theorem 21

Let f : [ a , b ] → R + be a * differentiable mapping on [ a , b ] with a < b . If f ∗ is multiplicatively s-convex on [ a , b ] , then the following midpoint-type inequality holds:

(7) f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( f ∗ ( a ) ) f ∗ a + b 2 2 ( s + 1 ) ( f ∗ ( b ) ) b − a 4 ( s + 1 ) ( s + 2 ) ≤ ( ( f ∗ ( a ) ) ( f ∗ ( b ) ) ) ( 2 2 − s + 1 ) ( b − a ) 4 ( s + 1 ) ( s + 2 ) .

Proof

From Lemma 13 and properties of * integral, we have

f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ exp b − a 4 ∫ 0 1 ln f ∗ ( 1 − t ) a + t a + b 2 t d t × exp b − a 4 ∫ 0 1 ln f ∗ ( 1 − t ) a + b 2 + t b t − 1 d t = exp b − a 4 ∫ 0 1 t ln f ∗ ( 1 − t ) a + t a + b 2 d t × exp b − a 4 ∫ 0 1 ( 1 − t ) ln f ∗ ( 1 − t ) a + b 2 + t b d t .

By using the multiplicative s -convexity of f ∗ , we have

f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ exp b − a 4 ∫ 0 1 t ln ( f ∗ ( a ) ) ( 1 − t ) s f ∗ a + b 2 t s d t × exp b − a 4 ∫ 0 1 ( 1 − t ) ln f ∗ a + b 2 ( 1 − t ) s ( f ∗ ( b ) ) t s d t = exp b − a 4 ∫ 0 1 t ( 1 − t ) s ln ( f ∗ ( a ) ) + t s + 1 ln f ∗ a + b 2 d t × exp b − a 4 ∫ 0 1 ( 1 − t ) s + 1 ln f ∗ a + b 2 + t s ( 1 − t ) ln ( f ∗ ( b ) ) d t = exp b − a 4 ln ( f ∗ ( a ) ) ∫ 0 1 t ( 1 − t ) s d t + ln f ∗ a + b 2 ∫ 0 1 t s + 1 d t × exp b − a 4 ln f ∗ a + b 2 ∫ 0 1 ( 1 − t ) s + 1 d t + ln ( f ∗ ( b ) ) ∫ 0 1 t s ( 1 − t ) d t = exp b − a 4 1 ( s + 1 ) ( s + 2 ) ln ( f ∗ ( a ) ) + 1 s + 2 ln f ∗ a + b 2 × exp b − a 4 1 s + 2 ln f ∗ a + b 2 + 1 ( s + 1 ) ( s + 2 ) ln ( f ∗ ( b ) ) .

Thus we have

(8) f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( f ∗ ( a ) ) 1 ( s + 1 ) ( s + 2 ) f ∗ a + b 2 2 s + 2 ( f ∗ ( b ) ) 1 ( s + 1 ) ( s + 2 ) b − a 4 = ( f ∗ ( a ) ) f ∗ a + b 2 2 ( s + 1 ) ( f ∗ ( b ) ) b − a 4 ( s + 1 ) ( s + 2 )

and the first inequality in (7) is proved. To prove the second inequality in (7), by using the multiplicative s -convexity of f ∗ , we obtain

(9) f ∗ a + b 2 2 s − 1 ≤ ( ( f ∗ ( a ) ) ( f ∗ ( b ) ) ) 1 s + 1 .

A combination of (8) and (9) gives the inequality

f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( f ∗ ( a ) ) f ∗ a + b 2 2 ( s + 1 ) ( f ∗ ( b ) ) b − a 4 ( s + 1 ) ( s + 2 ) ≤ ( f ∗ ( a ) ) ( ( f ∗ ( a ) ) ( f ∗ ( b ) ) ) 1 s + 1 2 ( s + 1 ) 2 1 − s ( f ∗ ( b ) ) b − a 4 ( s + 1 ) ( s + 2 ) = ( ( f ∗ ( a ) ) ( f ∗ ( b ) ) ) ( 2 2 − s + 1 ) ( b − a ) 4 ( s + 1 ) ( s + 2 ) ,

which proves the second inequality in (7).

So, the proof is completed.□

Corollary 22

In Theorem 7, if we take s = 1 , then we obtain

f a + b 2 ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( f ∗ ( a ) ) f ∗ a + b 2 4 ( f ∗ ( b ) ) b − a 4 ( s + 1 ) ( s + 2 ) ≤ ( ( f ∗ ( a ) ) ( f ∗ ( b ) ) ) ( b − a ) 8 .

Theorem 23

Let f : [ a , b ] → R + be a * differentiable mapping on [ a , b ] with a < b . If f ∗ is multiplicatively s-convex on [ a , b ] , then the following trapezoid-type inequality holds:

G ( f ( a ) , f ( b ) ) ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( f ∗ ( a ) f ∗ ( b ) ) s . 2 s + 1 2 s + 1 ( s + 1 ) ( s + 2 ) ( b − a ) ,

where G is the geometric mean.

Proof

From Lemma 14, properties of * integral and the multiplicative s -convexity of f ∗ , we have

G ( f ( a ) , f ( b ) ) ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ exp ( b − a ) ∫ 0 1 ln f ∗ ( ( 1 − t ) a + t b ) 1 2 − t d t = exp ( b − a ) ∫ 0 1 t − 1 2 ln ∣ f ∗ ( ( 1 − t ) a + t b ) ∣ d t = exp ( b − a ) ∫ 0 1 t − 1 2 ln ( f ∗ ( a ) ) ( 1 − t ) s ( f ∗ ( b ) ) t s d t = exp ( b − a ) ∫ 0 1 t − 1 2 ( ( 1 − t ) s ln ( f ∗ ( a ) ) + t s ln ( f ∗ ( b ) ) ) d t = exp ( b − a ) ln ( f ∗ ( a ) ) ∫ 0 1 t − 1 2 ( 1 − t ) s d t + ln ( f ∗ ( b ) ) ∫ 0 1 t − 1 2 t s d t

= exp ( b − a ) ln ( f ∗ ( a ) ) ∫ 0 1 t − 1 2 ( 1 − t ) s d t exp b − a 2 ln ( f ∗ ( b ) ) ∫ 0 1 t − 1 2 t s d t = exp ( b − a ) s 2 s + 1 2 s + 1 ( s + 1 ) ( s + 2 ) ln ( f ∗ ( a ) ) exp ( b − a ) s 2 s + 1 2 s + 1 ( s + 1 ) ( s + 2 ) ln ( f ∗ ( b ) ) = ( f ∗ ( a ) f ∗ ( b ) ) s . 2 s + 1 2 s + 1 ( s + 1 ) ( s + 2 ) ( b − a ) .

So, the proof is completed.□

Corollary 24

In Theorem 23, if s = 1 , then we have

G ( f ( a ) , f ( b ) ) ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ ( f ∗ ( a ) f ∗ ( b ) ) b − a 8 ,

which is established by Khan and Budak in [54].

Corollary 25

In Corollary 24, if f ∗ ≤ M , then we obtain

G ( f ( a ) , f ( b ) ) ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b ≤ M b − a 4 .

4 Applications to special means

We shall consider the means for arbitrary real numbers a , b .

  1. The Arithmetic mean: A ( a 1 , a 2 ) = a 1 + a 2 2 .

  2. The Harmonic mean: H ( a 1 , a 2 ) = 2 a 1 a 2 a 1 + a 2 , a 1 , a 2 > 0 .

  3. The Logarithmic mean: L ( a 1 , a 2 ) = a 2 − a 1 ln a 2 − ln a 1 , a 1 , a 2 > 0 , and a 1 ≠ a 2 .

  4. The p -Logarithmic mean: L p ( a 1 , a 2 ) = a 2 p + 1 − a 1 p + 1 ( p + 1 ) ( a 2 − a 1 ) 1 p , a 1 , a 2 > 0 , a 1 ≠ a 2 and p ∈ R \ { − 1 , 0 } .

Proposition 26

Let a , b ∈ R with 0 < a < b . Then, we have

e A p ( a , b ) − L p p ( a , b ) ≤ ( e a p − 1 + b p − 1 ) 5 p ( b − a ) 72 .

Proof

The assertion follows from Corollary 18, applied to the function f ( t ) = e t p with p ≥ 2 whose f ∗ ( t ) = e p t p − 1 and ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b = exp ( − L p p ( a , b ) ) . □

Proposition 27

Let a , b ∈ R with 0 < a < b and r > 0 . Then, we have

e H − 1 ( a , b ) − L − 1 ( a , b ) ≤ e − ( b − a ) 4 r 2 .

Proof

The assertion follows from Corollary 25, applied to the function f ( t ) = e 1 t whose f ∗ ( t ) = e − 1 t 2 , M = e − 1 r 2 and ∫ a b ( f ( ϰ ) ) d ϰ 1 a − b = exp ( − L − 1 ( a , b ) ) . □

5 Conclusion

In this study, first, a new identity for multiplicatively differentiable functions is presented. Using this identity, a new Simpson-type integral inequality for multiplicatively s -convex functions is obtained. Additionally, new integral inequalities of the midpoint and trapezoid-types for multiplicative integrals are derived. It is shown that some of our results improve upon those in the existing literature, while others are generalizations of known results. Applications of these results to special means are also presented. The findings of this study may inspire further research by scholars in this field.

Acknowledgement

I would like to express my deepest gratitude to the referees and the handling editor for their valuable comments.

  1. Funding information: The author states that no funding was involved.

  2. Author contributions: The author has accepted responsibility for the entire content of this article and approved its submission.

  3. Conflict of interest: The author states that there is no conflict of interest.

  4. Ethical approval: The conducted research is not related to either human or animal use.

  5. Data availability statement: No data were used to support the study.

References

[1] P. Agarwal, S. S. Dragomir, M. Jleli and B. Samet, Advances in Mathematical Inequalities and Applications, Springer, Singapore, 2018. 10.1007/978-981-13-3013-1Suche in Google Scholar

[2] H. Budak, M. A. Ali, and M. Tarhanacı, Some new quantum Hermite-Hadamard-like inequalities for coordinated convex functions, J. Optimization Theory Appl. 186 (2020), no.3, 899–910, DOI: https://doi.org/10.1007/s10957-020-01726-6. 10.1007/s10957-020-01726-6Suche in Google Scholar

[3] S. I. Butt, M. K. Bakula, D. Pečarić and J. Pečarić, Jensen-Grüss inequality and its applications for the Zipf-Mandelbrot law, Math. Methods Appl. Sci. 44 (2021), no. 2, 1664–1673, DOI: https://doi.org/10.1002/mma.6869. 10.1002/mma.6869Suche in Google Scholar

[4] S. I. Butt, A. O. Akdemir, M. Nadeem, N. Mlaiki, İİssscan, and T. Abdeljawad, (m,n)-Harmonically polynomial convex functions and some Hadamard type inequalities on the co-ordinates, AIMS Math. 6 (2021), no.5, 4677–4690, DOI: https://doi.org/10.3934/math.2021275. 10.3934/math.2021275Suche in Google Scholar

[5] H. Hudzik and L. Maligranda, Some remarks on s-convex functions, Aequationes Math. 48 (1994), 100–111. 10.1007/BF01837981Suche in Google Scholar

[6] İ. İşcan, Hermite-Hadamard type inequalities for harmonically convex functions, Hacettepe 43 (2014), no. 6, 935–942, DOI: https://doi.org/10.48550/arXiv.1303.6089. 10.15672/HJMS.2014437519Suche in Google Scholar

[7] M. Kadakal and İ. İşcan, Exponential type convexity and some related inequalities, J. Ineq. Appl. 2020 (2020), 82, DOI: https://doi.org/10.1186/s13660-020-02349-1. 10.1186/s13660-020-02349-1Suche in Google Scholar

[8] M. Kadakal, İ. İşcan, and H. Kadakal, Semi P-geometric-arithmetically functions and some new related inequalities, Filomat 37 (2023), no.21, 7017–7028, DOI: https://doi.org/10.2298/FIL2321017K. 10.2298/FIL2321017KSuche in Google Scholar

[9] H. Kadakal, H. Maden, M. Kadakal, and İ. İşcan, Some new integral inequalities for n-times differentiable r-convex and r-concave functions, Miskolc Math. Notes. 20 (2019), no. 2, 997–1011, DOI: https://doi.org/10.18514/MMN.2019.2489. 10.18514/MMN.2019.2489Suche in Google Scholar

[10] M. A. Latif and T. Du, Hermite-Hadamard type inequalities for harmonically-convex functions using fuzzy integrals, Filomat 36 (2022), no. 12, 4099–4110, DOI: https://doi.org/10.2298/FIL2212099L. 10.2298/FIL2212099LSuche in Google Scholar

[11] M. A. Noor, K. I. Noor, and S. Iftikhar, Some characterizations of harmonic convex functions, Int. J. Anal. Appl. 15 (2017), no. 2, 179–187, DOI: https://doi.org/10.28924/2291-8639-15-2017-179. 10.28924/2291-8639-15-2017-179Suche in Google Scholar

[12] S. Numan and İ. İşcan, On exponential type P-functions, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 70 (2021), no. 1, 497–509, DOI: https://doi.org/10.31801/cfsuasmas.745357. 10.31801/cfsuasmas.745357Suche in Google Scholar

[13] S. Ozcan, Hermite-hadamard type inequalities for multiplicatively s-convexfunctions, Cumhuriyet Sci. J. 41 (2020), no. 1, 245–259, DOI: http://dx.doi.org/10.17776/csj.663559. 10.17776/csj.663559Suche in Google Scholar

[14] S. Özcan, Hermite-Hadamard type inequalities for exponentially p-convex stochastic processes, Sakarya Univ. J. Sci. 23 (2019), no. 5, 1012–1018, DOI: https://doi.org/10.16984/saufenbilder.561040. 10.16984/saufenbilder.561040Suche in Google Scholar

[15] S. Özcan, On refinements of some integral inequalities for differentiable prequasiinvex functions, Filomat 33 (2019), no. 14, 4377–4385, DOI: https://doi.org/10.2298/FIL1914377O. 10.2298/FIL1914377OSuche in Google Scholar

[16] S. Özcan, Some integral inequalities of Hermite-Hadamard type for multiplicatively s-preinvex functions, Int. J. Math. Model. Comput. 9 (2019), no. 4, 253–266. Suche in Google Scholar

[17] T. Rasheed, S. I. Butt, D. Pečarić, and J. Pečarić, Generalized cyclic Jensen and information inequalities, Chaos Solitons Fractals 163 (2022), 112602, DOI: https://doi.org/10.1016/j.chaos.2022.112602. 10.1016/j.chaos.2022.112602Suche in Google Scholar

[18] E. Set, A. Gözpınar, and S. I. Butt, A study on Hermite-Hadamard-type inequalities via new fractional conformable integrals, Asian-Europian J. Math. 14 (2021), no. 2, 2150016, DOI: https://doi.org/10.1142/S1793557121500169. 10.1142/S1793557121500169Suche in Google Scholar

[19] E. Ünlüyol and Y. Erdaş, A generalization of Hermite-Hadamard-Fejer type inequalities for the p-convex function via α-generator, J. Math. 2023 (2023), 1185960, DOI: https://doi.org/10.1155/2023/1185960. 10.1155/2023/1185960Suche in Google Scholar

[20] S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-Hadamard Inequalities and Applications, RGMIA Monographs, Victoria University, 2000. Suche in Google Scholar

[21] J. Hadamard, Etude sur les proprietes des fonctions entieres en particulier daune fonction consideree par Riemann, J. Math. Pures Appl. 58 (1893), 171–215. Suche in Google Scholar

[22] J. E. Pečarić, F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings and Statistical Applications, Academic Press, Boston, 1992. Suche in Google Scholar

[23] S. I. Butt, M. Tariq, A. Aslam, H. Ahmad, and T. A. Nofal, Hermite-Hadamard type inequalities via generalized harmonic exponential convexity and applications, J. Func. Spaces 2021 (2021), 5533491, DOI: https://doi.org/10.1155/2021/5533491. 10.1155/2021/5533491Suche in Google Scholar

[24] H. Kadakal, Hermite-Hadamard type inequalities for two times differentiable arithmetic-harmonically convex functions, Cumhuriyet Sci. J. 40 (2019), no. 3, 670–678, DOI: https://doi.org/10.17776/csj.544984. 10.17776/csj.544984Suche in Google Scholar

[25] M. Kunt and İ. İşcan, Hermite-Hadamard-Fejer type inequalities for p-convex functions, Arab. J. Math. Sci. 23 (2017), no. 2, 215–230, DOI: https://doi.org/10.1016/j.ajmsc.2016.11.001. 10.1016/j.ajmsc.2016.11.001Suche in Google Scholar

[26] M. A. Latif, Inequalities of Hermite-Hadamard type for functions whose derivatives in absolute value are convex with applications, Arab J. Math. Sci. 21 (2015), no. 1, 84–97, DOI: https://doi.org/10.1016/j.ajmsc.2014.01.002. 10.1016/j.ajmsc.2014.01.002Suche in Google Scholar

[27] M. A. Latif, S. S. Dragomir, and E. Momoniat, On Hermite-Hadamard type integral inequalities for n-times differentiable m- and (α,m)-logarithmically convex functions, Filomat 30 (2016), no. 11, 3101–3114, DOI: https://doi.org/10.2298/FIL1611101L. 10.2298/FIL1611101LSuche in Google Scholar

[28] S. Maden, H. Kadakal, M. Kadakal, and İ. İşcan, Some new integral inequalities for n-time differentiable convex and concave functions, J. Nonlinear Sci. Appl. 10 (2017), 6141–6148, DOI: http://dx.doi.org/10.22436/jnsa.010.12.01. 10.22436/jnsa.010.12.01Suche in Google Scholar

[29] S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-convex functions, Konuralp J. Math. 8 (2020), no. 1, 158–164. 10.1186/s13660-020-02442-5Suche in Google Scholar

[30] S. Özcan, Hermite-Hadamard type inequalities for multiplicatively h-preinvex functions, Turkish J. Anal. Number Theory 9 (2020), no. 3, 65–70, DOI: https://doi.org/10.12691/tjant-9-3-5. 10.12691/tjant-9-3-5Suche in Google Scholar

[31] M. Z. Sarikaya and H. Budak, On generalized Hermite-Hadamard inequality for generalized convex function, Int. J. Nonlinear Anal. Appl. 8 (2017), no. 2, 209–222, DOI: https://doi.org/10.22075/IJNAA.2017.11313.1552. Suche in Google Scholar

[32] Y. Shuang and F. Qi, Integral inequalities of Hermite-Hadamard type for extended s-convex functions and applications, Mathematics 6 (2018), no. 11, 223, DOI: https://doi.org/10.3390/math6110223. 10.3390/math6110223Suche in Google Scholar

[33] B.-Y. Xi and F. Qi, Some integral inequalities of Hermite-Hadamard type for s-logarithmically convex functions, Acta Math. Sci. English Series 35A (2015), no. 3, 515–526, DOI: https://doi.org/10.13140/RG.2.1.4385.9044. Suche in Google Scholar

[34] S. S. Dragomir, R. P. Agarwal, and P. Cerone, On Simpsonas inequality and applications, J. Ineq. Appl. 5 (2000), no. 6, 533–579, DOI: https://doi.org/10.1155/S102558340000031X. 10.1155/S102558340000031XSuche in Google Scholar

[35] T. Abdeljawad, S. Rashid, Z. Hammouch, İ. İşcan, and Y.-M. Chu, Some new Simpson-type inequalities for generalized p-convex function on fractal sets with applications, Adv. Differential Equations 2020 (2020), 496. DOI: https://doi.org/10.1186/s13662-020-02955-9. 10.1186/s13662-020-02955-9Suche in Google Scholar

[36] M. A. Ali, H. Kara, J. Tariboon, S. Asawasamrit, H. Budak, and F. Hezenci, Some new Simpsonas-formula-type inequalities for twice-differentiable convex functions via generalized fractional operators, Symmetry 13 (2021), no. 12, 2249, DOI: https://doi.org/10.3390/sym13122249. 10.3390/sym13122249Suche in Google Scholar

[37] M. A. Ali, M. Abbas, H. Budak, P. Agarwal, G. Murtaza, and Y. M. Chu, New quantum boundaries for quantum Simpsonas and quantum Newtonas type inequalities for preinvex functions, Adv. Differential Equations. 64 (2021), 1–21, DOI: https://doi.org/10.1186/s13662-021-03226-x. 10.1186/s13662-021-03226-xSuche in Google Scholar

[38] M. A. Ali, H. Budak, M. Z. Sarıkaya, and Z. Zhang, Ostrowski and Simpson type inequalities for multiplicative integrals, Proyecciones 40 (2021), no. 3, 743–763, DOI: http://dx.doi.org/10.22199/issn.0717-6279-4136. 10.22199/issn.0717-6279-4136Suche in Google Scholar

[39] M. Alomari, M. Darus, and S. S. Dragomir, New inequalities of Simpsonas type for s-convex functions with applications, RGMIA Res. Rep. Coll. 12 (2009), no. 4, 9. Suche in Google Scholar

[40] S. Chasreechai, M. A. Ali, S. Naowarat, T. Sitthiwirattham, and K. Nonlaopon, On some Simpson’s and Newton’s type of inequalities in multiplicative calculus with applications, AIMS Math. 8 (2023), no. 2, 3885–3896, DOI: https://doi.org/10.3934/math.2023193. 10.3934/math.2023193Suche in Google Scholar

[41] K.-C. Hsu, S.-R. Hwang, and K.-L. Tseng, Some extended Simpson-type inequalities and applications, Bull. Iran. Math. Soc. 43 (2017), no. 2, 409–425. Suche in Google Scholar

[42] A. Kashuri, B. Meftah, and P. O. Mohammed, Some weighted Simpson type inequalities for differentiable s-convex functions and their applications, J. Frac. Calc. Nonlinear Sys. 1 (2021), no. 1, 75–94, DOI: https://doi.org/10.48185/jfcns.v1i1.150. 10.48185/jfcns.v1i1.150Suche in Google Scholar

[43] C.-Y. Luo, T.-S. Du, M. Kunt, and Y. Zhang, Certain new bounds considering the weighted Simpson-like type inequality and applications, J. Ineq. Appl. 2018 (2018), 332, DOI: https://doi.org/10.1186/s13660-018-1924-3. 10.1186/s13660-018-1924-3Suche in Google Scholar PubMed PubMed Central

[44] M. A. Noor, K. I. Noor, and M. U. Awan, Simpson-type inequalities for geometrically relative convex functions, Ukr. Math. J. 70 (2018), 1145–1154, DOI: https://doi.org/10.1007/s11253-018-1558-0. 10.1007/s11253-018-1558-0Suche in Google Scholar

[45] J. Pecaric and S. Varosanec, A note on Simpsonas inequality for functions of bounded variation, Tamkang J. Math. 31 (2000), no. 3, 239–242, DOI: https://doi.org/10.5556/j.tkjm.31.2000.398. 10.5556/j.tkjm.31.2000.398Suche in Google Scholar

[46] M. Z. Sarıkaya, E. Set, and M. E. Özdemir, On new inequalities of Simpsonas type for s-convex functions, Computers Math. Appl. 60 (2010), no. 8, 2191–2199, DOI: https://doi.org/10.1016/j.camwa.2010.07.033. 10.1016/j.camwa.2010.07.033Suche in Google Scholar

[47] X. You, M. A. Ali, H. Budak, H. Kara, and D. Zhao, Some parameterized Simpsonas type inequalities for differentiable convex functions involving generalized fractional integrals, Adv. Continuous Discrete Models 2022 (2022), 22, DOI: https://doi.org/10.1186/s13662-022-03696-7. 10.1186/s13662-022-03696-7Suche in Google Scholar

[48] M. A. Ali, M. Abbas, Z. Zhang, I. B. Sial, and R. Arif, On integral inequalities for product and quotient of two multiplicatively convex functions, Asian Res. J. Math. 12 (2019), no. 3, 1–11, DOI: https://doi.org/10.9734/arjom/2019/v12i330084. 10.9734/arjom/2019/v12i330084Suche in Google Scholar

[49] M. A. Ali, M. Abbas, and A. A. Zafer, On some Hermite-Hadamard integral inequalities in multiplicative calculus, J. Ineq. Special Func., 10 (2019), no. 1, 111–122. Suche in Google Scholar

[50] A. Berhail and B. Meftah, Midpoint and trapezoid type inequalities for multiplicatively convex functions, [Submitted on 31 Jul 2022] DOI: https://doi.org/10.48550/arXiv.2208.00535. Suche in Google Scholar

[51] H. Budak and K. Özcelik, On Hermite-Hadamard type inequalities for multiplicative fractional integrals, Miskolc Math. Notes 21 (2020), no. 1, 91–99, DOI: https://doi.org/10.18514/MMN.2020.3129. 10.18514/MMN.2020.3129Suche in Google Scholar

[52] H. Kadakal, Multiplicatively P-functions and some new inequalities, New Trends Math. Sci. 6 (2018), no. 4, 111–118, DOI: https://doi.org/10.20852/ntmsci.2018.321. 10.20852/ntmsci.2018.321Suche in Google Scholar

[53] H. Kadakal and M. Kadakal, Multiplicatively preinvex P-functions, J. Sci. Arts 23 (2023), no. 1, 21–32, DOI: https://doi.org/10.46939/J.Sci.Arts-23.1-a02. 10.46939/J.Sci.Arts-23.1-a02Suche in Google Scholar

[54] S. Khan and H. Budak, On midpoint and trapezoid type inequalities for multiplicative integrals, Mathematica 64 (2022), 87, no. 1, 95–108, DOI: https://doi.org/10.24193/mathcluj.2022.1.11. 10.24193/mathcluj.2022.1.11Suche in Google Scholar

[55] M. A. Noor, F. Qi, and M. U. Awan, Some Hermite-Hadamard type inequalities for log-h-convex functions, Analysis 33 (2013), no. 4, 1–9, DOI: https://doi.org/10.1524/anly.2013.1223. 10.1524/anly.2013.1223Suche in Google Scholar

[56] S. Özcan, Some integral inequalities of Hermite-Hadamard type for multiplicatively preinvex functions, AIMS Math. 5 (2020), no. 2, 1505–1518, DOI: https://doi.org/10.3934/math.2020103. 10.3934/math.2020103Suche in Google Scholar

[57] S. Özcan, Hermite-Hadamard type inequalities for exponential type multiplicatively convex functions, Filomat 37 (2023), no. 28, 9777–9789, DOI: https://doi.org/10.2298/FIL2328777O. 10.2298/FIL2328777OSuche in Google Scholar

[58] M. Grossman and R. Katz, Non-Newtonian Calculus, Lee Press, Pigeon Cove, MA, 1972. Suche in Google Scholar

[59] A. E. Bashirov, E. M. Kurpınar and A. Özyapıcı, Multiplicative calculus and applications, J. Math. Anal. Appl. 337 (2008), no. 1, 36–48, DOI: https://doi.org/10.1016/j.jmaa.2007.03.081. 10.1016/j.jmaa.2007.03.081Suche in Google Scholar

[60] A. E. Bashirov and M. Rıza, On complex multiplicative differentiation, TWMS J. Appl. Eng. Math. 1 (2011), no. 1, 75–85. Suche in Google Scholar

[61] Y. L. Daletskii and N. I. Teterina, Multiplicative stochastic integrals, Uspekhi Matematicheskikh Nauk 27 (1972), 2:164, 167–168. Suche in Google Scholar

[62] A. E. Bashirov, E. Kurpınar, Y. Tandogdu, and A. Özyapıcı, On modeling with multiplicative differential equations, Appl. Math. A J. Chinese Univ. 26 (2011), no. 4, 425–438, DOI: https://doi.org/10.1007/s11766-011-2767-6. 10.1007/s11766-011-2767-6Suche in Google Scholar

[63] R. L. Karandikar, Multiplicative decomposition of non-singular matrix valued continuous semimartingales, Ann Probability 10 (1982), no. 4, 1088–1091. 10.1214/aop/1176993734Suche in Google Scholar

[64] E. Mısırlı and A. Özyapıcı, Exponential approximations on multiplicative Calculus, Proc. Jangjeon Math. Soc. 12 (2009), no. 2, 227–236. Suche in Google Scholar

[65] M. Rıza, A. Özyapıcı, and E. Kurpınar, Multiplicative finite difference methods, Quart. Appl. Math. 67 (2009), no. 4, 745–754. 10.1090/S0033-569X-09-01158-2Suche in Google Scholar
Received: 2023-08-17
Revised: 2024-03-17
Accepted: 2024-05-24
Published Online: 2025-02-22

© 2025 the author(s), published by De Gruyter

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https://doi.org/10.1515/dema-2024-0060
Schlagwörter für diesen Artikel
multiplicative calculus; multiplicatively s-convex function; Simpson inequality; midpoint-type inequality; trapezoid-type inequality
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  1. Research Articles
  2. On approximation by Stancu variant of Bernstein-Durrmeyer-type operators in movable compact disks
  3. Circular n,m-rung orthopair fuzzy sets and their applications in multicriteria decision-making
  4. Grand Triebel-Lizorkin-Morrey spaces
  5. Coefficient estimates and Fekete-Szegö problem for some classes of univalent functions generalized to a complex order
  6. Proofs of two conjectures involving sums of normalized Narayana numbers
  7. On the Laguerre polynomial approximation errors and lower type of entire functions of irregular growth
  8. New convolutions for the Hartley integral transform imbedded in the Banach algebras and convolution-type integral equations
  9. Some inequalities for rational function with prescribed poles and restricted zeros
  10. Lucas difference sequence spaces defined by Orlicz function in 2-normed spaces
  11. Evaluating the efficacy of fuzzy Bayesian networks for financial risk assessment
  12. Fixed point results for contractions of polynomial type
  13. Estimation for spatial semi-functional partial linear regression model with missing response at random
  14. Investigating the controllability of differential systems with nonlinear fractional delays, characterized by the order 0 < η ≤ 1 < ζ ≤ 2
  15. New forms of bilateral inequalities for K-g-frames
  16. Rate of pole detection using Padé approximants to polynomial expansions
  17. Existence results for nonhomogeneous Choquard equation involving p-biharmonic operator and critical growth
  18. Note on the shape-preservation of a new class of Kantorovich-type operators via divided differences
  19. Geršhgorin-type theorems for Z1-eigenvalues of tensors with applications
  20. New topologies derived from the old one via operators
  21. Blow up solutions for two-dimensional semilinear elliptic problem of Liouville type with nonlinear gradient terms
  22. Infinitely many normalized solutions for Schrödinger equations with local sublinear nonlinearity
  23. Nonparametric expectile shortfall regression for functional data
  24. Advancing analytical solutions: Novel wave insights and methodologies for beta fractional Kuralay-II equations
  25. A generalized p-Laplacian problem with parameters
  26. A study of solutions for several classes of systems of complex nonlinear partial differential difference equations in ℂ2
  27. Towards finding equalities involving mixed products of the Moore-Penrose and group inverses by matrix rank methodology
  28. ω ℒ -biprojective and ω ¯ -contractible Banach algebras
  29. Coefficient functionals for Sakaguchi-type-Starlike functions subordinated to the three-leaf function
  30. Solutions of several general quadratic partial differential-difference equations in ℂ2
  31. Inequalities for the generalized trigonometric functions with respect to weighted power mean
  32. Optimization of Lagrange problem with higher-order differential inclusion and special boundary-value conditions
  33. Hankel determinants for q-starlike functions connected with q-sine function
  34. System of partial differential hemivariational inequalities involving nonlocal boundary conditions
  35. A new family of multivalent functions defined by certain forms of the quantum integral operator
  36. A matrix approach to compare BLUEs under a linear regression model and its two competing restricted models with applications
  37. Weighted composition operators on bicomplex Lorentz spaces with their characterization and properties
  38. Behavior of spatial curves under different transformations in Euclidean 4-space
  39. Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
  40. A new kind of Durrmeyer-Stancu-type operators
  41. A study of generalized Mittag-Leffler-type function of arbitrary order
  42. On the approximation of Kantorovich-type Szàsz-Charlier operators
  43. Split quaternion Fourier transforms for two-dimensional real invariant field
  44. Quantum injectivity of G-frames in Hilbert spaces
  45. Some results on disjointly weakly compact sets
  46. On Motzkin sequence spaces via q-analog and compact operators
  47. Existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in ℝ3
  48. Stability analysis of linear time-invariant difference-differential system with constant and distributed delays
  49. The discriminant of quasi m-boundary singularities
  50. Norm constrained empirical portfolio optimization with stochastic dominance: Robust optimization non-asymptotics
  51. Fuzzy stability of multi-additive mappings
  52. On inequalities involving n-polynomial exponential-type convex functions
  53. Review Article
  54. Characterization generalized derivations of tensor products of nonassociative algebras
  55. Special Issue on Differential Equations and Numerical Analysis - Part II
  56. Existence and optimal control of Hilfer fractional evolution equations
  57. Persistence of a unique periodic wave train in convecting shallow water fluid
  58. Existence results for critical growth Kohn-Laplace equations with jumping nonlinearities
  59. Monotonicity and oscillation for fractional differential equations with Riemann-Liouville derivatives
  60. Nontrivial solutions for a generalized poly-Laplacian system on finite graphs
  61. Stability and bifurcation analysis of a modified chemostat model
  62. Special Issue on Nonlinear Evolution Equations and Their Applications - Part II
  63. Analytic solutions of a generalized complex multi-dimensional system with fractional order
  64. Extraction of soliton solutions and Painlevé test for fractional Peyrard-Bishop DNA model
  65. Special Issue on Recent Developments in Fixed-Point Theory and Applications - Part II
  66. Some fixed point results with the vector degree of nondensifiability in generalized Banach spaces and application on coupled Caputo fractional delay differential equations
  67. On the sum form functional equation related to diversity index
  68. Special Issue on International E-Conference on Mathematical and Statistical Sciences - Part II
  69. Simpson, midpoint, and trapezoid-type inequalities for multiplicatively s-convex functions
  70. Converses of nabla Pachpatte-type dynamic inequalities on arbitrary time scales
  71. Special Issue on Blow-up Phenomena in Nonlinear Equations of Mathematical Physics - Part II
  72. Energy decay of a coupled system involving a biharmonic Schrödinger equation with an internal fractional damping
  73. Special Issue on Some Integral Inequalities, Integral Equations, and Applications - Part II
  74. Nonlinear heat equation with viscoelastic term: Global existence and blowup in finite time
  75. New Jensen's bounds for HA-convex mappings with applications to Shannon entropy
  76. Special Issue on Approximation Theory and Special Functions 2024 conference
  77. Ulam-type stability for Caputo-type fractional delay differential equations
  78. Faster approximation to multivariate functions by combined Bernstein-Taylor operators
  79. (λ, ψ)-Bernstein-Kantorovich operators
  80. Some special functions and cylindrical diffusion equation on α-time scale
  81. (q, p)-Mixing Bloch maps
  82. Orthogonalizing q-Bernoulli polynomials
  83. On better approximation order for the max-product Meyer-König and Zeller operator
  84. Moment-based approximation for a renewal reward process with generalized gamma-distributed interference of chance
  85. A note on linear compositions of the Mellin convolution operators in the weighted Mellin-Lebesgue spaces
  86. A new perspective on generalized Laguerre polynomials
  87. Global existence of semilinear system of Klein-Gordon equations in anti-de Sitter spacetime
  88. Estimates for Durrmeyer-type exponential sampling series in Mellin-Orlicz spaces
  89. ℐ-αβ-statistical relative uniform convergence for double sequences and its applications
  90. New developments for the Jacobi polynomials
  91. Generalization of Sheffer-λ polynomials
  92. Special Issue on Variational Methods and Nonlinear PDEs
  93. A note on mean field type equations
  94. Ground states for fractional Kirchhoff double-phase problem with logarithmic nonlinearity
  95. Solution of nonlinear Langevin equations involving Hilfer-Hadamard fractional order derivatives and variable coefficients
  96. Bifurcation, quasi-periodic, and wave solutions to the fractional model of optical fibers in communication systems
  97. Multiplicity and concentration behavior of solutions for the generalized quasilinear Schrödinger equation with critical growth
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Heruntergeladen am 20.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/dema-2024-0060/html
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