Generalized notion of integral inequalities of variables

• Mashael M. AlBaidani , Abdul Hamid Ganie and Asia Fahd Mohammad Almuteb
From the journal Open Physics

Abstract

The fractional structures of variables using Riemann–Liouville notion have been analyzed by various authors. The novel idea of this article is to introduce the new notion of weighted behavior on random variables using integral inequalities. In view of these, we obtain some new generalized fractional inequalities by using this new fractional integration of continuous random variables.

1 Introduction

In recent times, the integral inequalities play a vital role in science, engineering, and technology. The authors are much attracted to study and analyze the structure in the fields like physics, statistics, biology, chemistry, and engineering as witnessed in refs. [1,2,3] and many others. The basic concern of fractional calculus has a straight knock on the solution of various problems of fast growing sciences to stimulate much interest in such field and to show its visibility in them. The different types of procedures and applications of fractional derivatives have been developed as can be found in refs. [4,5, 6,7,8, 9,10]. In this direction, Riemann–Liouville (RL) and Grunwald–Letnikov are most notable authors. It was Leonhard Euler in around 1720s gave the concept of how to extend the factorial to noninteger values. This developed a rich theory utilizing over the scientific world.

For a real number , the Euler gamma function Γ ( ) for { 0 , 1 , 2 , 3 , } can be expressed as an improper integral:

Γ ( ) = 0 e t t 1 d t .

It is observed that

1. Γ ( + 1 ) = ! for N ,

2. Γ ( + 1 ) = Γ ( ) for R { 0 , 1 , 2 , 3 , } .

It was Grunwald–Letnikow who forwarded the definition of fractional derivative order y as follows:

(1) y [ ξ ( t ) ] = lim h 0 1 h y r = 0 γ ( y , r ) ξ ( t + ( h r ) h ) ,

(2) γ ( y , r ) = ( 1 ) r Γ ( y + 1 ) r ! Γ ( y r + 1 ) ,

where Γ is the gamma function and h is the time increment as can be found in refs. [11, 12,13,14], and many others.

The Riemann–Liouville fractional integral of order η 0 as can be seen in refs. [15,16,17] is defined as follows:

(3) 0 y = 0 t g ( τ ) d τ

and

(4) g ( τ ) = 1 Γ ( y + 1 ) [ f ( τ ) ( t τ ) y ] .

For 0 < y < 1 , we see from (2) that

(5) γ ( y , 0 ) = 1

and

(6) r = 1 γ ( y , r ) = 1 .

Using the notion of theory of probability concept, we have following:

Using (5) the “present” (i.e., ξ ( 0 ) ) is noticed in (1) having one as probability.

The classical approach of the Riemann–Liouville fractional derivative was reformulated by Caputo and he then gave the solution of fractional differential equations under given initial conditions. It was Grunwald–Letnikov who later put forward the fractional calculus given by Leibnitz in a new way. Importance of fractional calculus by using its execution to real-world issues in applied analyses, statistics, physics, and fluid mechanics eta cetera can be witnessed in refs. [18,19,20, 21,22,23, 24,25,26, 27,28,29] and many others and the classical approach was generalized.

Now as in refs. [30,31, 32,33,34, 35,36,37, 38,39], we have following definitions in an extended sense:

Definition 1.1

The left and right Riemann–Liouville fractional integral of order κ 0 , respectively, are given by

(7) I α κ + = 1 Γ ( κ ) α u ( u t ) κ 1 F ( t ) d t for u > α ,

and

(8) I β κ = 1 Γ ( κ ) u β ( u t ) κ 1 F ( t ) d t for u < β ,

where Γ ( κ ) = 0 e v v κ 1 d v is called as the Gamma function and I α 0 + = I α 0 + F ( u ) = F ( u ) .

For κ 0 and 0 , we have following well-known results:

(9) I α κ + I α + = I α κ + + I α κ + I α + = I α + I α κ + .

Also, as in refs. [30,35], and many others, we have following definitions:

Definition 1.2

For 1 p < , the space L p , k ( α , β ) of real-valued Lebesgue measurable functions F on [ α , β ] such that

(10) F L p , k ( α , β ) = α β F ( v ) p v k d v 1 p < ,

for κ 0 .

Definition 1.3

For c R and 1 p < , we consider the space X c p ( α , β ) of all real-valued Lebesgue measurable functions F on [ α , β ] such that

(11) F X c p = α β v c F ( v ) p d v v 1 p < ,

and choosing p = , we define it as follows:

(12) F X c = ess sup α v β v c F ( v ) .

Remark 1.4

By choosing c = ( k + 1 ) p with 1 p < , k 0 , then X c p ( α , β ) gets coincide to L p , k ( α , β ) -space, and when we assume c = 1 p for 1 p < then space X c p ( α , β ) reduced to the classical space L p ( α , β ) .

Definition 1.5

Let F L 1 , s and k 0 , the generalized Riemann–Liouville fractional integrals + I a κ , s and I a κ , s with order κ 0 are given by

(13) I α κ , s + = ( s + 1 ) 1 κ Γ ( κ ) α v ( v s + 1 t s + 1 ) κ 1 t s F ( t ) d t for v > a .

(14) I β κ , s = ( s + 1 ) 1 κ Γ ( κ ) v β ( v s + 1 t s + 1 ) κ 1 t s F ( t ) d t for β > v .

It is observed that the integral formulas given by (13) and (14) are, respectively, known as right generalized Riemann–Liouville integral and left generalized Riemann–Liouville fractional integral.

Throughout the article, random variable of X will be abbreviated by r.v. and : [ α , β ] R + as a positive continuous function.

Following the authors [10,11,12, 13,14,15, 16,36,37, 38,39,40], we have following generalized definitions:

Definition 1.6

Let X be a r.v. having a positive p.d.f. F given on [ α , β ] . Then, for s 0 and α < ς β , the -weighted fractional expectation function of order κ is defined as follows:

(15) E X , κ , ( ς ) = I α κ , s + [ ς F ( ς ) ] = ( s + 1 ) 1 κ Γ ( κ ) α ς ( ς s + 1 t s + 1 ) κ 1 t s + 1 ( t ) F ( t ) d t .

Definition 1.7

We define the -weighted fractional expectation for a r.v. X E ( X ) of order κ as follows:

(16) E X E ( X ) , κ , ( ς ) = ( s + 1 ) 1 κ Γ ( κ ) α ς ( ς s + 1 t s + 1 ) κ 1 × ( t E ( X ) ) ( t ) t s F ( t ) d t .

Note by choosing ς = β , the aforementioned definitions reduces to following:

Definition 1.8

Let X be a r.v. with a positive p.d.f. F defined on [ α , β ] . Then, for s 0 , we define the -weighted fractional expectation function of order κ as follows:

(17) E X , κ , , ( ς ) = ( s + 1 ) 1 κ Γ ( κ ) α β ( β s + 1 t s + 1 ) κ 1 × t s + 1 ( t ) F ( t ) d t .

Definition 1.9

For mathematical expectation E ( X ) of r.v. X having a positive p.d.f. F . Then, on [ α , β ] , s 0 and α < ς β , the -weighted fractional variance function of order κ is defined as follows:

(18) σ X , κ , 2 ( ς ) = I α κ , s + [ ( ς E ( X ) ) 2 F ( ς ) ] = ( s + 1 ) 1 κ Γ ( κ ) α ς ( ς s + 1 t s + 1 ) κ 1 ( t E ( X ) ) 2 t s × ( t ) F ( t ) d t .

Definition 1.10

If ς = β , then -weighted fractional variance function of order κ is given as follows:

(19) σ X , κ , 2 ( ς ) = ( s + 1 ) 1 κ Γ ( κ ) α β ( β s + 1 t s + 1 ) κ 1 × ( t E ( X ) ) 2 t s ( t ) F ( t ) d t .

We have following important points:

Remark 1.11

(D1): Choosing s = 0 and κ = 1 and ( t ) = 1 for every t [ α , β ] in Definition 1.6, we get E X , 1 , 1 = = E ( X ) as the classical expectation of r.v. X .

(D2): Setting s = 0 and κ = 1 and ( t ) = 1 for every t [ α , β ] in Definition 1.8, we get σ X , 1 , 1 2 = σ 2 ( X ) = α β ( t E ( X ) ) 2 F ( t ) d t as classical variance of r.v. X .

(D3): Setting κ = 1 and ( t ) = 1 for every t [ α , β ] , we get the well-known result I κ [ F ( β ) ] = 1 .

2 Main results

This section will be concerning the new generalization of outcomes of continuous r.v. having fractional integral order.

Theorem 2.1

Let X be a r.v. with p.d.f. F : [ α , β ] R + . Then for each α < ς β , κ 0 and s 0 , we see

(i) the inequality

(20) I α κ , s + [ F ( ς ) ] σ X , κ , 2 ( ς ) ( E X E ( X ) , κ , ( ς ) ) 2 F 2 ( s + 1 ) 1 κ ( ς s + 1 α s + 1 ) κ Γ ( κ + 1 ) I α κ , s + [ ( ς ) ς 2 s + 2 ] ( I α κ , s + [ ( ς ) ς ] ) 2

holds provided F L [ α , β ] and

(ii) the inequality

(21) I α κ , s + [ F ( ς ) ] σ X , κ , 2 ( ς ) ( E X E ( X ) , κ , ( ς ) ) 2 1 2 ( ς s + 1 α s + 1 ) 2 ( I α κ , s + [ ( ς ) ς ] ) 2

holds.

Proof

Define a function Q for t , m ( α , ς ) , α < ς β as follows:

(22) Q ( t , m ) = ( Q 1 ( t ) Q 1 ( m ) ) ( Q 2 ( t ) Q 2 ( m ) ) ,

where κ 0 .

Now multiplying (22) by ( ς s + 1 t s + 1 ) κ 1 Γ ( κ ) t s p ( t ) both sides, where the p is a function p : [ α , β ] R + with t ( α , ς ) , and then integrating the resulting identity from α to ς and have

(23) ( s + 1 ) 1 κ Γ ( κ ) α ς ( ς s + 1 t s + 1 ) κ 1 p ( t ) Q ( t , m ) t s d t = I α , κ , s + [ p Q 1 Q 2 ( ς ) ] Q 2 ( m ) I α , κ , s + [ p Q 1 ( ς ) ] Q 1 ( m ) I α , κ , s + [ p Q 2 ( ς ) ] + Q 1 ( m ) Q 2 ( m ) I α , κ , s + [ p ( ς ) ] .

Now multiplying (23) by ( ς s + 1 m s + 1 ) κ 1 Γ ( κ ) p ( m ) m s for m ( α , ς ) , and then integrating the resulting identity over ( α , ς ) with respect to m , we see

(24) ( s + 1 ) 2 2 κ Γ 2 ( κ ) α ς α ς ( ς s + 1 t s + 1 ) κ 1 × ( ς s + 1 m s + 1 ) κ 1 p ( t ) p ( m ) Q ( t , m ) t s m s d t d m = 2 I α , κ , s + [ p ( ς ) ] I α , κ , s + [ p Q 1 Q 2 ( ς ) ] 2 I α , κ , s + [ p Q 1 ( ς ) ] I α , κ , s + [ p Q 2 ( ς ) ] .

Now in (24), choosing p ( ς ) = ( ς ) F ( ς ) and Q 1 ( ς ) = Q 2 ( ς ) = ς s + 1 E ( X ) , ς ( α , β ) , we see

(25) ( s + 1 ) 2 2 κ Γ 2 ( κ ) α ς α ς ( ς s + 1 t s + 1 ) κ 1 ( ς s + 1 m s + 1 ) κ 1 × ( t ) F ( t ) ( m ) F ( m ) ( t s + 1 m s + 1 ) 2 t s m s d t d m = 2 I α κ , s + [ F ( ς ) ] I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] 2 [ I α κ , s + F ( ς ) ( ς s + 1 E ( X ) ) 2 ] .

But, on the other hand, we see

(26) ( s + 1 ) 2 2 κ Γ 2 ( κ ) α ς α ς ( ς s + 1 t s + 1 ) κ 1 ( ς s + 1 m s + 1 ) κ 1 × F ( t ) ( t ) F ( m ) ( m ) ( t s + 1 m s + 1 ) 2 t s m s d t d m F 2 2 ( s + 1 ) 1 κ ( ς s + 1 α s + 1 ) κ Γ ( κ + 1 ) I α κ , s + [ ( ς ) ς 2 s + 2 ] 2 ( I α κ , s + [ ( ς ) ς ] ) 2 ) .

Consequently, using definitions 1.6, 1.7, and 1.9, the part (i) of the result follows from (25) and (26).

(ii) We have

(27) ( s + 1 ) 2 2 κ Γ 2 ( κ ) α ς α ς ( ς s + 1 t s + 1 ) κ 1 ( ς s + 1 m s + 1 ) κ 1 × F ( t ) F ( m ) ς ( t ) ς ( m ) ( t s + 1 m s + 1 ) 2 t s m s d t d m sup t , m [ α , t ] ( t s + 1 m s + 1 ) 2 ( I α κ , s + [ F ( ς ) ] ) 2 = ( t s + 1 α s + 1 ) 2 ( I α κ , s + [ F ( ς ) ] ) 2 .

Consequently, from (25) and (27), we get (21) as desired.□

Corollary 2.2

If ( t ) = 1 for every t [ α , β ] and κ 0 , the continuous r.v. X having p.d.f. F in [ α , β ] , then,

(i) the inequality

(28) β s + 1 α s + 1 Γ ( κ ) σ X , κ , 2 E X , κ 2 F 2 ( β s + 1 α s + 1 ) 2 κ + 2 Γ ( κ + 1 ) Γ ( κ + 3 ) ( β s + 1 α s + 1 ) κ + 1 Γ ( κ + 1 ) 2

gets satisfied if s 0 and F L [ α , β ] ;

(ii) the inequality

(29) β s + 1 α s + 1 Γ ( κ ) σ X , κ , 2 E X , κ 2 1 2 ( β s + 1 α s + 1 ) 2 κ Γ 2 ( κ )

gets satisfied for any s 0 .

Deduction 2.3

The first part of Theorem 1 in ref. [20] will be deduced if we set κ = 1 , ( t ) = 1 for every t [ α , β ] and s = 0 in (i) of Corollary 2.2.

Deduction 2.4

The last part of Theorem 1 in ref. [20] will be deduced if we set κ = 1 , ( t ) = 1 for every t [ α , β ] and s = 0 in (ii) of Corollary 2.2.

Theorem 2.5

For a r.v. X having a p.d.f. F : [ α , β ] R + . Then

(i) for all α < ς β , κ 0 , 0 and s 0 , we have

(30) I α κ , s + [ F ( ς ) ] σ X , 2 ( ς ) + I α , s + [ F ( ς ) ] σ X , κ , 2 ( ς ) 2 ( E X E ( X ) , κ , ( ς ) ) ( E X E ( X ) , ( ς ) ) F 2 ( s + 1 ) 1 κ ( ς s + 1 α s + 1 ) κ Γ ( κ + 1 ) I α , s + [ ς 2 s + 2 ] + F 2 ( s + 1 ) 1 ( ς s + 1 α s + 1 ) Γ ( + 1 ) I α κ , s + [ ς 2 s + 2 ] 2 ( I α κ , s + [ ς ] ) ( I α , s + [ ς ] ) )

holds for F L [ α , β ] and

(ii) the inequality

(31) I α κ , s + [ F ( ς ) ] σ X , 2 ( ς ) + I α , s + [ F ( ς ) ] σ X , κ , 2 ( ς ) 2 ( E X E ( X ) , κ , ( ς ) ) ( E X E ( X ) , ( ς ) ) ( ς s + 1 α s + 1 ) ( I α κ , s + [ ς ] ) ( I α , s + [ ς ] )

holds for any α < ς β , κ 0 , 0 and s 0 .

Proof

From (22), we can write

(32) ( s + 1 ) 2 κ Γ ( κ ) Γ ( ) α ς α ς ( ς s + 1 t s + 1 ) κ 1 ( ς s + 1 m s + 1 ) κ 1 × p ( t ) p ( m ) Q ( t , m ) t s m s d t d m = I α , κ , s + [ p ( ς ) ] I α , , s + [ p Q 1 H 2 ( ς ) ] + I α , , s + [ p ( ς ) ] I α , κ , s + [ p Q 1 H 2 ( ς ) ] I α , κ , s + [ p H 2 ( ς ) ] I α , , s + [ p Q 1 ( ς ) ] I α , , s + [ p H 2 ( ς ) ] I α , κ , s + [ p Q 1 ( ς ) ] .

For ς ( α , β ) , put p ( ς ) = ( ς ) F ( ς ) , Q 1 ( ς ) = Q 2 ( ς ) = ς s + 1 E ( X ) in (32), we see

(33) ( s + 1 ) 2 κ Γ ( κ ) Γ ( ) α ς α ς ( ς s + 1 t s + 1 ) κ 1 ( ς s + 1 m s + 1 ) κ 1 × ( t ) F ( t ) ( m ) F ( m ) ( t s + 1 m s + 1 ) 2 t s m s d t d m = I α κ , s + [ F ( ς ) ] I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] + I α , s + [ F ( ς ) ] I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] 2 I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] I α , s + [ F ( ς ) ( t s + 1 E ( X ) ) ] .

But we also see

(34) ( s + 1 ) 2 κ Γ ( κ ) Γ ( ) α ς α ς ( ς s + 1 t s + 1 ) κ 1 ( ς s + 1 m s + 1 ) κ 1 × ( t ) F ( t ) ( m ) F ( m ) ( t s + 1 m s + 1 ) 2 t s m s d t d m F 2 ( s + 1 ) 1 κ ( ς s + 1 α s + 1 ) κ Γ ( κ + 1 ) I α , s + [ ς 2 s + 2 ] + ( s + 1 ) 1 ( ς s + 1 α s + 1 ) Γ ( + 1 ) I α κ , s + [ ς 2 s + 2 ] 2 ( I α κ , s + [ ς ] ) ( I α , s + [ ς ] ) ] .

Consequently, using this and by involving (33) will establish part (i) of the result.

Now to establish (ii), we shall make use of following

sup t , m [ α , ς ] ( t s + 1 m s + 1 ) 2 = ( ς s + 1 α s + 1 ) 2

and get

( s + 1 ) 2 κ Γ ( κ ) Γ ( ) α ς α ς ( ς s + 1 t s + 1 ) κ 1 × ( t ) g ( t ) ( m ) g ( m ) ( t s + 1 m s + 1 ) 2 t s m s d t d m ( ς s + 1 α s + 1 ) 2 ( I α κ , s + [ ς ] ) ( I α , s + [ ς ] ) .

Thus, using this equation and applying (33) will prove part (ii), i.e., (31) is proved.□

Theorem 2.6

Let X be a r.v. having p.d.f. g : [ α , β ] R + . Then

(35) I α κ , s + [ F ( ς ) ] σ X , κ , 2 ( ς ) ( E X E ( X ) , κ , ( ς ) ) 2 1 4 ( β s + 1 α s + 1 ) 2 ( I α κ , s + [ ς ] ) 2

for every α < ς β , κ 0 and s 0 .

Proof

From Theorem 3.1 of ref. [40], one can write

(36) 0 I α κ , s + [ p ( ς ) ] I α κ , s + [ p Q 1 2 ( ς ) ] ( I α κ , s + [ p Q 1 ( ς ) ] ) 2 1 4 ( I α κ , s + [ p ( ς ) ] ) 2 ( m ) 2 .

For ς [ α , β ] , we choose p ( ς ) = F ( ς ) and Q 1 ( ς ) = ς s + 1 E ( X ) , then = β s + 1 E ( X ) and m = α s + 1 E ( X ) . Thus, from (36), we can have

(37) 0 I α κ , s + [ F ( ς ) ] I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] ( I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] ) 2 1 4 ( I α κ , s + [ F ( ς ) ] ) 2 ( β s + 1 α s + 1 ) 2 .

This yields that

(38) I α κ , s + [ F ( ς ) ] σ X , κ , 2 ( ς ) ( E X E ( X ) , κ , ( ς ) ) 2 1 4 ( β s + 1 α s + 1 ) 2 ( I α κ , s + [ ς ] ) 2 .

This completes the proof.□

Now choosing ς = β and = 1 , we have following corollary:

Corollary 2.7

For a p.d.f. g of r.v. X , we have

( β s + 1 α s + 1 ) κ 1 Γ ( κ ) σ X , κ 2 ( ς ) ( E X E ( X ) , κ ( ς ) ) 2 1 4 Γ 2 ( κ ) ( β s + 1 α s + 1 ) 2 κ

for κ 0 and s 0 .

Deduction 2.8

The Theorem 3.7 of ref. [22] will be deduced if we set s = 1 and ( t ) = 1 for every t [ α , β ] in Corollary 2.7.

Theorem 2.9

For a r.v. X with p.d.f. g : [ α , β ] R + . Then

(39) I α κ , s + [ F ( ς ) ] σ X , 2 ( ς ) + I α , s + [ F ( ς ) ] σ X , κ , 2 ( ς ) + 2 ( α s + 1 E ( X ) ) ( β s + 1 E ( X ) ) × I α κ , s + [ F ( ς ) ] I α , s + [ F ( ς ) ] ( α s + 1 + β s + 1 2 E ( X ) ) ( I α κ , s + [ F ( ς ) ] ( E X E ( X ) , ( ς ) ) + I α β , s + [ F ( ς ) ] ( E X E ( X ) , α ( ς ) ) )

for every α < ς β , κ 0 , 0 and s 0 .

Proof

From Theorem 3.4 of ref. [40], we can write

(40) [ I α κ , s + [ p ( ς ) ] I α , s + [ p Q 1 2 ( ς ) ] + [ I α , s + [ p ( ς ) ] I α κ , s + [ p Q 1 2 ( ς ) ] 2 I α κ , s + [ p Q 1 ( ς ) ] I α , s + [ p Q 1 ( ς ) ] ] 2 [ ( I α κ , s + [ p ( ς ) ] I α κ , s + [ p Q 1 ( ς ) ] ) ( I α , s + [ p Q 1 ( ς ) ] m I α , s + [ p ( ς ) ] ) + ( I α , s + [ p Q 1 ( ς ) ] m I α , s + [ p ( ς ) ] ) ( I α , s + [ p ( ς ) ] I α , s + [ p Q 1 ( ς ) ] ) ] 2 .

Choosing p ( ς ) = F ( ς ) and Q 1 ( ς ) = ς s + 1 E ( X ) in (40) yields

(41) [ I α κ , s + [ F ( ς ) ] I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] + I α , s + [ F ( ς ) ] I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] 2 I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] ] 2 [ I α κ , s + [ F ( ς ) ] I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] × ( I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] m I α , s + [ F ( ς ) ] ) × ( I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] m I α κ , s + [ F ( ς ) ] ) × ( M I α , s + [ F ( ς ) ] I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] ) ] 2 .

Now from (33) and (41) and using the fact that the left hand side of (33) is positive, we see

(42) I α κ , s + [ F ( ς ) ] I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] + I α , s + [ F ( ς ) ] I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] 2 I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] I α κ , s + [ F ( ς ) ] I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] × ( I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] m I α , s + [ F ( ς ) ] ) × ( I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] m I α κ , s + [ F ( ς ) ] ) × ( M I α , s + [ F ( ς ) ] I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) ] ) .

This yields us

I α κ , s + [ F ( ς ) ] I α , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] + I α , s + [ F ( ς ) ] I α κ , s + [ F ( ς ) ( ς s + 1 E ( X ) ) 2 ] ( I α κ , s + [ F ( ς ) ] ( E X E ( X ) , ( ς ) ) + I α , s + [ F ( ς ) ( E X E ( X ) , κ , ( ς ) ) ] ) + m ( I α κ , s + [ F ( ς ) ] ( E X E ( X ) , ( ς ) ) + I α , s + [ F ( ς ) ( E X E ( X ) , κ , ( ς ) ) ] ) .

Consequently, the result is established by simple calculation with utilizing the values of and m from Theorem 2.6.□

3 Conclusion

In this paper, we have presented various concepts of fractional calculus. Also, some new generalizations of outcomes for continuous random variables having fractional order have been given. Furthermore, certain definitions like ℧-weighted fractional expectation and variance, have been introduced and their various properties have been studied. Moreover, new bounds and inequalities have been established. The consequences of the results obtained in this manuscript are more general and extensive than the pre-existing known results.

Acknowledgments

The authors are thankful to the anonymous reviewers for their valuable comments and suggestions towards the improvement of the article.

1. Funding information: The authors state no funding involved.

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

3. Conflict of interest: The authors state no conflict of interest.

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