SOME CHEBYSHEV TYPE INEQUALITIES FOR GENERALIZED RIEMANN-LIOUVILLE OPERATOR Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 12. № 2 (2020). С. 87-95.

SOME CHEBYSHEV TYPE INEQUALITIES FOR GENERALIZED RIEMANN-LIOUVILLE OPERATOR

B. HALIM, A.SENOUCI, M. SOFRANI

Abstract. In this paper we are interested in the famous inequality introduced by Chebvshev. This inequality has several generalizations and applications in different fields of mathematics and others. In particular it is important for us the applications of fractional calculus for the different integral Chebvshev type inequalities.

We establish and prove some theorems and corollaries relating to fractional integral, by applying more general fractional integral operator than Riemann-Liouville one:

v(x)

Щ

(x -t)

a—1

ß-1

f (t)u(t)dt, x> 0

where a > 0 ft ^ 1, u and v locally integrable non-negative weight functions, r is the Euler Gamma-function. First, fractional integral Chebvshev type inequalities are obtained for operator K™',v with two synchronous or two asynchronous functions and by induction for several functions. Second, we consider an extended Chebvshev functional

b b b b T(f,g,p, q) ■= q(x)dx p(x)f (x)g(x)dx + p(x)dx q(x)f(x)g(x)dx

} \( }

/ q(x)f(x)dx\ \ / p(x)g(x)dx

\a / \a

where p, q are positive integrable weight functions on [a, b]. In this case fractional integral weighted inequalities are established for two fractional integral operators Kul',v! and Ku2',v2, with two synchronous or asynchronous functions, where ai = a2, fti = P2 and u\ = u2, v\ = v2- In addition, a fractional integral Holder type inequality for several functions is established using the operator K™'^■ At the end, another fractional integral Chebvshev

Keywords: Chebvshev functional, Integral Inequalities, RL-fractional operator. Mathematics Subject Classification: 34B45, 81Q15

X

B. IIai.ixi. A.Senouci, M. Sofrani, Some Chebyshev type inequalities for generalized Riemann-Liouville operator.

© Halim B.,Senouci A., Sofrani M. 2020.

This paper is supported by University of Tiaret, PRFU project, code COOL03UN140120180002. Submitted September 30, 2019.

1. Introduction Let 0 ^ a <b < ro ^d g be two integrable funct ions on [a, b] and

T (f,g) := J f (x)g{x)dx - - ^^ I J f (x)dx\ I J g{x)dx | . (1.1)

a \a / \a

The Chebyshev functional (1,1) has many applications in numerical quadrature, transform theory, probability, study of existence of solutions of differential equations and in statistical problems. The following inequality called Chebyshev integral inequality is well known, see [3].

Lemma 1.1. If f and g are two synchronous funcHons on [a,b], i.e

(f (r) - f (p))(g(r) - g(p)) > 0),

for each, r,p E [a,b], then

T(f,g) > 0. (1.2)

If f,g are asynchronous on [a,b], i.e.,

(f (r) - f (p))(g(r) - g(p)) < 0),

for each, r,p E [a,b], then inequality (1.2) is reversed. The constant -- is the best possible in

b - a

inequality (1.2).

We consider the extended Chebyshev functional defined as follows

b b

T (f,g,p,q) := J q(x)dx J p(x)f (x)g(x)dx

a a

b b

+ J p(x)dx J q(x)f (x)g(x)dx

a a

i \( f

/ q(x)f (x)dx / p(x)g(x)dx

\a / \a

b \ / b

J q(z)g(x)dx I ,

where p, q are positive integrable weight functions on [a, b].

If q(x) = p(x),x E [a,b], in T(f,g,p,q), we have the following lemma, see [7].

Lemma 1.2. If f and g are two synchronous functions on [a,b], then b b b b T(f,g,p) := Jp(x)dx Jp(x)f (x)g(x)dx — Jp(x)f (x)dx Jp(x)g(x)dx ^ 0. (1.3)

a a a a

If f,9 are asynchronous [a, b], then inequality (1.3) is reversed.

Remark 1.1. If p(x) = 1, in (1.3) we obtain the classical Chebychev inequality. In the following we give some basic definitions.

Definition 1.1. For 1 ^ p < to we denote by Lp := Lp(0, to) the set of all Lebesgue measurable functions f such that

|f{x)\pdxj < to.

Definition 1.2. Let z > 0,r,s > 0. The gamma and the beta functions are defined as follows

T(z) = j tz-le-tdt z > 0, 0

i

B(r, s) = J tr-1(1 - t)s-1dt. 0

Definition 1.3. The Riemann-Liouville fractional integral operators of order a ^ 0 of function f(x) E L1[a, b], —to < a < b < +to, are defined by

X

3aa+f (x) = ra) /(x — t)a-lf № x > a.

a

b

JU(x) = f(t — x)a-1 f(t)dt, x<b.

r(a) J

X

For a = 0 we denote J%+ by Ja.

Definition 1.4. A real valued function f : [0, to) ^ R is said to be in the space C^, ^ E R, if there exists a real number p > ^ such that f(x) = xp f1(x), where f1 E C[0, to).

The following theorems were proved in [5].

Theorem 1.1. Let f and g be two synchronous functions on (0, to). Then for all

> 0, a > 0,

Ja(fg)(t) > ^a+^jy(t)Jag(t) (i.4)

for all t > 0, a > 0. The inequality (1.4) is reversed if the functions are asynchronous on (0, to).

Theorem 1.2. Let f and g be two synchronous functions on (0, to). Then

j-a ffi

JP(f9)(t) + w7TT-rvJa(f9)(t) > Jaf (t)J"9(t) + J"f (t)Ja9(t). (1.5)

r(a + 1) r(/ + 1)

for allt > 0, a > 0, / > 0. The inequality (1.5) is reversed if the functions are asynchronous on (0, to).

L, p' ^ 1 such that 11 1

and Lp', respectively. Then

Theorem 1.3. Let p ^ 1, p' ^ 1 such that ^ + 1 = 1, if f and g are two functions in Lp

Ja(f9)(x) ^ (Jafp(x))> (Jagp'(x))^

for all x > 0, a > 0.

The following theorems were proved in [2].

Theorem 1.4. Let [fi}l^i^n be n positive increasing functions on (0, ro) then

J a

(i=n \ i=n

n fi) (x) > (J a(l)(x)f-n) n J aMx) i=l J i=l

for all x > 0, a > 0.

Theorem 1.5. Let f and g be two functions defined on (0, ro), such that f is increasing and g is differentiable and there exists a real number m := inf^o g'(x). Then

Ja(fg)(x) > (Ja(l))-1 Jaf (x)Jag(x) -^-JV(x) + mJa(xf)(x)

a + 1

is valid for all x > 0, a > 0.

2. Main Results

The aim of this work is to extend the results of [2] and [5] to a more general fractional integral operator, than the Riemann-Liouville one.

Definition 2.1. Let a > 0, 0 ^ 1, 1 ^ p < ro and the integral operator be defined as

X

K«f f (X) = YfáJ(x - f)a-1 [ln (J)]"-1 f (t)u(t)dt, x > 0 (2.1)

0

defined from Lp to Lp space, with locally integrable non-negative weight functions u and v.

We mention that for a > 0,0 ^ 1 necessary and sufficient conditions for the boundedness, see [6, Thm. 3.1], and compactness, see [6, Thm. 4.1], of the integral operator K" from Lp to Lq spaces with 0 < p,q < ro were found for locally integrable non-negative weight functions

U, V.

Remark 2.1. If v(x) = u(x) = 1,0 =1, the operator K"'1 coincides with the classical Riemann-Liouville fractional integral operator.

To simplify the calculations, we denote

K := , k(x, t) := (x - t)a-1 ln^-^= 0. Then the integral operator in inequality (2.1) becomes

K f (x) = ^r)ík(x, t)f(t)u(t)dt, x> 0.

r(a) J

0

Theorem 2.1. Let f, g be two synchronous functions on (0, ro), u and v locally integrable non-negative weight functions. Then

K(fg)(x) > (K(1))-1 Kf (x)Kg(x), (2.2)

where

X

K(1)(x) = Í k(x, t)u(t)dt. T(a) J 0

Inequality (2.2) is reversed if the functions are asynchronous on (0, ro).

Proof. Since the functions f and g are synchronous on (0, to), then for all r ^ 0, p ^ 0 we have:

( f(r) — f(p)) (g(r) — g(p)) > 0.

Hence,

f(r)g(r) + ¡(p)g(p) > f(r)g(p) + f(r)g(p). (2.3)

v (x)

Multiplying both sides of inequality (2.3) by k(x, r)u(r), r E (0,x), we get:

r(a)

'^r^k^-i t) f(T)g(r)u(r) + ^X{k(x, r)f(p)g(p)u(r) ^^X{k(x, r)f(r)g(p)u(r) 1(a) 1(a) r(a)

+ W\k(x, r)f (P)^(r)u(r).

r(a)

(2.4)

Integrating inequality (2.4) with respect to r over (0,x), we obtain

X X

pOi f k(x, r)f(r)9(r)u(r)dr + f(p)9(p)Y;a)J k(x, r)u(r) dr 00

X X

> 9(P)Y(!))S Hx, T)f(T)u(T)dr+ 1(9)^0)1 k(x, r)f(r)u(r)dr. 00

This implies:

K( fg)(x) + f(p)g(p)K(1)(x) > g(p)K(f)(x) + f(p)K(g)(x). (2.5)

v (x)

Multiplying both sides of (2.5) by k(x, p)u(p), we get:

r(a)

^X{k(x, p)u(p)K(fg)(x) + ^X{k(x, p)u(p)f(p)g(p)K(1)(x) 1(a) 1(a)

^^г\k(x, P)u(P)9(P)K(f)(x) + Pr\k(x,P)u(P)f(P)K(9)(x). 1(a) 1(a)

We integrate the obtained inequality with respect to p over (0,x):

X X

K( f9)(x)^(x\ I k(x, p)u(p)dp + K(1)(x)^(x) / k(x, p)u(p)f(p)g(p)dp

r(a) J r(a) J

00

X X

> K f (x)^!4 / k(x,p)u(p)g(p)dp + K (x)V-(x\[ k(x,p)f(p)u(p)dp. r(a) J r(a) J

00

Hence,

K( f g)(x)K(1)(x) + K(1)(x)K( fg)(x) > K f (x)Kg(x) + Kg(x)K f (x).

This yields:

K( fg)(x) > (K(1))-1 Kf (x)Kg(x).

If f and g are asynchronous, the proof is similar to that of synchronous case. The proof is complete. □

Remark 2.2. Theorem 2.1 applied with v(x) = u(x) = 1, / =1 gives Theorem 1.1.

Theorem 2.2. Let {fi}l^i^n be n positive increasing functions on [0, ro) and u and v be locally integrable non-negative weight functions. Then

l=n

K n A (*) > (K(1)(x))(1-ra^Kh(x) \i=1 J i=1

for all x > 0.

Proof. We prove this Theorem by induction. We suppose that

fi=n-1\ i=n-1

m ^ (K(nm)(2-ra) | | k tax)

K I] A) (x) > (K(1)(x))(2-ra^ KU(x). (2.6)

i=i / i=i

ri=n— l

Since {are positive increasing functions, then YllJi h is an increasing function. Hence, we can apply Theorem 2.1 with J}f = g, fn = f, and we obtain

i=n— 1

(i=n \ /i=n-l \

n fi) (x) = K(fg)(x) > (K(1))-1 k( n fi) (x)Kfn(x).

Therefore, by (2.6), we get

(i=n \ / i=n \

K ( n fi) (*) ^ (K(1))-1 (K(1))2-ra n K fi) (x)K fn(x), \i=l ' \i=l '

and the proof is complete. □

Remark 2.3. Theorem 2.2 with v(x) = u(x) = 1,0 =1 gives Theorem 1.4.

Considering f\ = f, i = 1, 2,..., n, in Theorem 2.2, we get the following Corollary.

Corollary 2.1. Let f be an increasing positive function on (0, ro), u andv locally integrable non-negative weight functions, then

K( fn)(x) > (K(1)(x)f-n) (K f (x))n .

Now we consider the next two operators

X

Kl f(x) = r^y ( X - tr-l ln^-l (I) f(t)ul(t)dt, l0

X

K2 f (x) = ^J(X - tr-ln"2- (f ) f(t)U2(t)dt.

20

Theorem 2.3. Let f, g be two synchronous functions on (0, ro) p,q : [a, b] ^ (0, ro) be integrable, Ui and Vi i= 1, 2, locally integrable non-negative weight functions. Then

K2q(x)Kl(pfg)(x)+ Klp(x)K2(qf g)(x) > K2(qg)(x)Kl(pf)(x) + ^(qf)(x)Kl(pg)(x). (2.7)

for all x > 0. Inequality (2.7) is reversed if the functions are asynchronous on (0, ro).

Vl (x)

Proof. We multiply both sides of inequality (2.3) by kl(x, r)ul(r)p(r), tE (0,x), and

r(a)

integrating the resulting inequality with respect to r over (0,x), we find that

Kl(pf g)(x) + Kl(p)(x) f(p)g(p) > Kl(pf)(x)g(p) + Kl(pg)(x) f(p). (2.8)

V2 (x)

Again multiplying inequality (2.8) by k2(x, p)u2(p)q(p) and integrating the resulting inequality with respect to p over (0,x). This leads as to inequality (2.7). □

Letting q(x) = p(x) in Theorem 2.3, we get the following Corollary.

Corollary 2.2. Let f, g be two synchronous functions on [0, to), p : [a, b] ^ (0, to), uí be positive integrable weight functions and ví, i = 1, 2, be positive functions. Then

K2P(x)Ki(pfg)(x) + Kip(x)K2(pfg)(x) ^K2(pg)(x)Ki(p f)(x)

+ K2(pf)(x)Ki(pg)(x) ( . )

for all x > 0. Inequality (2.9) is reversed if the functions are asynchronous on (0, to).

Theorem 2.3 with Ki = K2 = K and q(x) = p(x) leads us to the following Corollary.

Corollary 2.3. Let f, g be two synchronous functions on (0, to), u andv be locally integrable non-negative weight functions. Then

Kp(x)K(pf g)(x) > K(pf)(x)K(pg)(x) (2.10)

for all x > 0. Inequality (2.10) is reversed if the functions are asynchronous on (0, to).

Theorem 2.3 with q(x) = p(x) = 1 gives the following corollary.

Corollary 2.4. Let f, g be two synchronous functions on (0, to), uí and vi,i = 1, 2 locally integrable non-negative weight functions. Then

K2(1)(x)Ki(fg)(x) + Ki(1)(x)K2(fg)(x) > K2g(x)KJ(x) + K2/(x)K^(x) (2.11)

for all x > 0. Inequality (2.11) is reversed if the functions are asynchronous on [0, to[.

If f = g in (2.11), we get the following corollary.

Corollary 2.5. Let f, f2 be positive and integrable functions on (0, to), and uí, u and vi, i = 1, 2, be locally integrable non-negative weight functions. Then

K2(1)Ki(f2)(x) + Ki(1)K2(f2)(x) ^ K2/(x)Kif (x)

for all x > 0.

Corollary 2.6. Let f be a positive and absolutely continuous function on (0, to) such that f ' > 0. Let uí and ví, i = 1, 2, be locally integrable non-negative weight functions. Then

K2(1)(x)Ki(f3)(x) + Ki(1)(x)K2(f3)(x) ^Ki(1)(x))-iK2f(x)(Kif (x))2

+ (K2(1)(x))-iKi/(x)(K2 f (x))2

for all x > 0.

Proof. We observe that the conditions f > 0, f ' > 0 imply that the functions f and f 2 are synchronous on (0, to). Hence, for all r, p > 0 we have

( f(r) - f(p))(f2(r) - f2(p)) > 0.

Therefore,

f(r) + f(p) > f(r)f 2(p) + f(p) f2(r).

Applying Theorem 2.1, we complete the proof. □

Remark 2.4. By applying Corollary 2.4 with ví(x) = ui(x) = 1, fa = 1, i = 1, 2, we arrive at Theorem 1.2.

Lemma 2.1. Suppose that j- + .... + j- = 1 forpí ^ 1 i = 1, 2,... ,n. If fí G LPi respectively,

In the following we shall make use a well known Holder inequality for many functions Lemma 2.1. Suppo then nr=1 fi E L1 and

c<

/ni fi\dx ^ [I I fi \Pidx J . (2.12)

Pi

0 i=1 i=1 \0

Theorem 2.4. Let pi ^ 1,i = 1, 2,... ,n such that

± + .... + ± = 1.

Pl Pa

If fi E LPi, u and v locally integrable non-negative weight functions, then

(i=n \ i=n

n fA (x) ^ n (Kft (x))« . (2.13)

i=l J i=l

for all x > 0.

Proof. For i = 1, 2,... ,n we consider the functions Fi, defined on (0,x) as follows

Fi(t) = k(x, i)« fi(t). By applying Holder's inequality, we obtain

v(x)

K(II /0(s)=fH/n fi(t)k(x, t)u(t)dt

i=l ( n i=l

( ) ~ t=n

V(X^ l[Fl(t)u(t)dt

0 i=l

T(a)

v(x)

Pi

(1(7) J F<' (Wdt

X 1 ni I k(x, t) ñ' (t)u(t)dt

n (K ff (*))

1

X))P' .

i=l

This proves inequality (2.13) and completes the proof. □

Remark 2.5. Theorem 2.4 applied with v(x) = u(x) = 1,0 = 1, n = 2 proves Theorem 1.3.

Theorem 2.5. Let f, g be two functions defined on (0, ro), u andv be locally integrable nonnegative weight functions. If f is increasing, g is differentiable and there exists a real number m := infecí g'(x), then

K(f g)(x) ^ (K(1))-1 K f (x)Kg(x) - m (K(1))-1 Kf (x)K(xd)(x) + mK(i/)(x)

holds for all x > 0, where id(x) = x.

Proof. We consider a function h(x) = g(x) - mx, where h is differentiable and increasing on [0, ro). Then f and h are synchronous on (0, ro). By applying Theorem 2.1, we conclude that

K ( f(x)(g - mx)) ^ (K(1))-1 Kf (x)K(g - mx).

Since K is linear, we have

K ( f(x)(g - mx)) = K (fg) (x) - mK(xf )(x).

This yields:

K(f g)(x) ^ (K(1))-1 K f (x)Kg(x) -m (K(1))-1 K(xd)(x)Kf (x)+mK(xf )(x). The proof is complete. □

X

Remark 2.6. By applying Theorem 2.5 for v(x) = u(x) = 1,P = 1, we obtain Theorem 1.5.

Theorem 2.1 applied to the decreasing functions f(x) and G(x) = g(x) — Mx for all x > 0, where M := sup^Q g'(x), gives rise to the following Corollary.

Corollary 2.7. Let f g be two functions defined on (0, to), u andv be locally integrable nonnegative weight functions. If f is decreasing, g is differentiable and there exists a real number M := supa,^Q g'(x), then

K( fg)(x) > (K(1))-1 K f (x)Kg(x) — M (K(1))-1 Kf (x)K(id)(x) + MK(xf )(x)

is valid for all x > 0.

We observe that our results generalize Theorems 1.1, 1.2, 1.3, 1.4 and 1.5.

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2. S. Belarbi and Z. Dahmani. On some new fractional integral inequality // J. Ineq. Pure Appl. Math. 10:3, 1-5 (2009).

3. P.L. Chebvshev. Sur les expressions approximatives des intégrales définies par les autres prises entre les mêmes limites // Proc. Math. Soc. Charkov 2, 93-98 (1882).

4. Z. Dahmani. Some results associate with fractional integrals involving the extended Chebyshev // Acta Univ. Apulensis Math. Inform. 27, 217-224 (2011).

5. Z. Dahmani. About some integral inequalities using Riemann-Liouville integrals // Gen. Math. 20:4, 63-69 (2012).

6. S.M. Farsani. On the boundedness and compactness of a certain integral operator // Banach J. Math. Anal. 7:2, 86-102 (2013).

7. D.S. Mitrinovic. Analytic inequalities. Springer, Berlin (1970).

Halim Benali

University of Tiaret

BP P 78 zaâroura

14000, Tiaret, Algeria

E-mail: benali .halim@univ-tiaret.dz

Senouci Abdelkader University of Tiaret BP P 78 zaâroura 14000, Tiaret, Algeria E-mail: kamer295@yahoo.fr

Sofrani Mohammed University of Tiaret BP P 78 zaâroura 14000, Tiaret, Algeria E-mail: aissamalik@yahoo.fr