Research article

On an integral and consequent fractional integral operators via generalized convexity

  • Received: 28 May 2020 Accepted: 27 July 2020 Published: 25 September 2020
  • MSC : 26D10, 31A10, 26A33

  • Fractional calculus operators are very useful in basic sciences and engineering. In this paper we study an integral operator which is directly related with many known fractional integral operators. A new generalized convexity namely exponentially (α, h?m)-convexity is defined which has been applied to obtain the bounds of unified integral operators. A generalized Hadamard inequality is established for the generalized convex functions. The established theorems reproduce several known results.

    Citation: Wenfeng He, Ghulam Farid, Kahkashan Mahreen, Moquddsa Zahra, Nana Chen. On an integral and consequent fractional integral operators via generalized convexity[J]. AIMS Mathematics, 2020, 5(6): 7632-7648. doi: 10.3934/math.2020488

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  • Fractional calculus operators are very useful in basic sciences and engineering. In this paper we study an integral operator which is directly related with many known fractional integral operators. A new generalized convexity namely exponentially (α, h?m)-convexity is defined which has been applied to obtain the bounds of unified integral operators. A generalized Hadamard inequality is established for the generalized convex functions. The established theorems reproduce several known results.


    Let h,q be integers with q>0. The classical Dedekind sum s(h,q) is defined by

    s(h,q)=q1a=1((aq))((haq)),

    where

    ((x))={x[x]12,ifxis not an integer;0,ifxis an integer

    and [x] is the largest integer not exceeding x. The Dedekind sum plays an important role in the Dedekind η function and has applications to many parts of mathematics (see [7,11,12]).

    For any nonnegative integer n, let Bn and Bn(X) be the n-th Bernoulli number and polynomial, respectively, which is defined by

    tet1=n=0Bntnn!, tetXet1=n=0Bn(X)tnn!,

    where ¯Bn(X)=Bn(X[X]) is the n-th periodic Bernoulli function in the interval (0,1] for n>1, ¯B1(X)=B1(X[X]) and if X is an integer, then ¯B1(X)=0. For positive integers m,n, we have the generalized Dedekind sum

    S(h,m,n,q)=q1a=1¯Bm(aq)¯Bn(ahq).

    Let p be an odd prime and let χ be any even Dirichlet character mod p. For any integers n,k, Xie and Zhang [8] showed that if n is an odd integer, then

    p1h=1χ(h)|S(h,n,n,p)|2k=2(n!)4k42(n1)k(pζ(2n)2π2n)2k|L(2nk,χ)|2ζ(4nk)+O(p2knexp(6lnplnlnp)), (1.1)

    they also have the following result for any even integer n

    p1h=1χ(h)|S(h,n,n,p)|2k=2(n!)4k42(n1)k(pζ(2n)2π2n)2k|L(2nk,χ)|2ζ(4nk)+O(p2k1). (1.2)

    In October 2000, Professor Todd Cochrane first introduced a sum analogous to the Dedekind sum as follows:

    C(h,q)=qa=1((ˉaq))((haq)),

    where ˉa satisfies aˉa1(modq), qa=1 denotes the summation over all 1aq such that (a,q)=1. Many scholars studied the properties of C(h,q). Zhang and Yi [13] gave the following upper bound estimate:

    |C(h,q)|q12d(q)ln2q,

    where d(q) is the divisor function. Ma et al. [4] gave the upper bound estimate of the incomplete Cochrane sum.

    Xu and Zhang [9] defined the high-dimensional Cochrane sum by the following equation:

    C(h,k,q)=qa1=1qak=1((a1q))((akq))((h¯a1akq)).

    For any fixed positive integer k with (q,k(k+1))=1, they gave the following upper bound estimate:

    |C(h,k,q)|2(k+1)2πk+1qk2d(q)(2k+2k)ω(q)lnk+1q,

    where ω(q) denotes the number of all different prime divisors of q. Liu [5] improved this result.

    For positive integers m,n, the main purpose of this paper is to study the generalized Cochrane sum with any Dirichlet character χ mod q as follows:

    C(h,m,n,q,χ)=qa=1χ(a)¯Bm(¯aq)¯Bn(ahq),

    which is an interesting generalization of Cochrane sum. Ren and Yi [6] studied the mean square value of C(h,1,1,p,χ), for a prime p1mod4 and the Legendre's symbol χmodp, they obtained

    p1h=1|C(h,1,1,p,χ)|2=1180p2p1A(p21+1p211)2+O(p1+o(1)), (1.3)

    where A is the set of quadratic residues of p, p1 is prime which is not equal to p. Liu and Zhang [2,3] studied the mean square value of C(h,m,n,q,χ) and its hybrid mean value formula when χ is the principal Dirichlet character. In this paper, by using an upper bound estimate of the Kloosterman sum with Dirichlet characters, we show that

    Theorem 1.1. Let p be an odd prime and let χ be any Dirichlet character mod p. For any integers m,n, we have

    C(h,m,n,p,χ)m!n!(2π)(m+n)p12ln2p,

    if χ(1)(1)n+m, then C(h,m,n,p,χ)=0.

    We also obtain the mean square value of C(h,m,n,p,χ) as follows:

    Theorem 1.2. Let p be an odd prime and let χ be any Dirichlet character mod p. For any integers m,n with χ(1)=(1)n+m, we have

    p1h=1|C(h,m,n,p,χ)|2=8(m!n!)2p2(2πi)2(m+n)ζ(2m)ζ(2n)ζ(2m+2n)|L(m+n,χ)|2+O(p2min(n,m)exp(4lnplnlnp)).

    If χ1 mod a prime p1mod4 is the Legendre's symbol, then according to the value of the Dirichlet L-function L(2,χ1), we also can get (1.3):

    Corollary 1.3. Let p1mod4 be a prime and the Legendre's symbol χ1 mod p. We have

    p1h=1|C(h,1,1,p,χ1)|2=1180p2p1A(p21+1p211)2+O(pexp(4lnplnlnp)),

    where A is the set of quadratic residues of p, p1 is prime, which is not equal to p.

    Moreover, we give the mean square value of S(h,m,n,p,χ) as follows:

    Theorem 1.4. Let p be an odd prime and let χ be any Dirichlet character mod p. For any integers m,n with χ(1)=(1)n+m, we have

    p1h=1χ(h)S(¯h,m,m,p)S(h,n,n,p)=8(m!n!)2p2(2πi)2(m+n)ζ(2m)ζ(2n)ζ(2m+2n)|L(m+n,χ)|2+O(p2min(n,m)exp(4lnplnlnp)).

    Obviously, Theorem 1.4 generalizes and improves (1.1) and (1.2) when k=1.

    To prove theorems, we need the following several lemmas.

    Lemma 2.1. Let p be an odd prime and an integer h with (h,p)=1, and let χ1 be any Dirichlet character mod p. For any positive integers m,n, if χ1(1)(1)m+n then C(h,m,n,p,χ1)=0, and if χ1(1)=(1)n+m, then we have

    C(h,m,n,p,χ1)=4m!n!(2πi)m+n1ϕ(p)χmodpχχ1(1)=(1)m¯χ(h)(+r=1G(χχ1,r)rm)(+s=1G(χ,s)sn),

    and

    C(h,m,n,p,χ1)=4m!n!(2πi)m+n1ϕ(p)χmodpχχ1(1)=(1)m¯χ(h)τ(χ)τ(χχ1)L(m,¯χχ1)L(n,¯χ),

    where τ(χ)=G(χ,1), G(χ,r)=p1a=1χ(a)e(rap) denotes Gauss sum, L(n,χ)=+t=1χ(t)tn is a Dirichlet L-function.

    Proof. Applying the orthogonality of multiplicative characters, it follows that

    C(h,m,n,p,χ1)=p1a=1χ1(a)¯Bm(¯ap)¯Bn(ahp)=1ϕ(p)χmodp{p1a=1χχ1(a)¯Bm(ap)}{p1b=1χ(b)¯Bn(hbp)}.

    Noting that [1]

    ¯Bn(x)=n!(2πi)n+r=r0e(xr)rn,

    and G(χ,hn)=¯χ(h)G(χ,n). We have

    C(h,m,n,p,χ1)=1ϕ(p)χmodp{p1a=1(m!(2πi)m+r=r0χχ1(a)e(rap)rm)}×{p1b=1(n!(2πi)n+s=s0χ(b)e(sbhp)sn)}=m!n!(2πi)m+n1ϕ(p)χmodp{+r=r01rmp1a=1χχ1(a)e(rap)}×{+s=s01snp1b=1χ(b)e(sbhp)}=m!n!(2πi)m+n1ϕ(p)χmodp{+r=r0G(χχ1,r)rm}{+s=s0G(χ,sh)sn}=m!n!(2πi)m+n1ϕ(p)χmodp¯χ(h)(1+χχ1(1)(1)m)(+r=1G(χχ1,r)rm)×(1+χ(1)(1)n)(+s=1G(χ,s)sn)=4m!n!(2πi)m+n1ϕ(p)χmodpχχ1(1)=(1)m¯χ(h)(+r=1G(χχ1,r)rm)(+s=1G(χ,s)sn),

    where χ1(1)=(1)m+n. If χ1(1)(1)m+n, then C(h,m,n,p,χ1)=0.

    Moreover, we also have

    C(h,m,n,p,χ1)=4m!n!(2πi)m+n1ϕ(p)χmodpχχ1(1)=(1)m¯χ(h)τ(χ)τ(χχ1)L(m,¯χχ1)L(n,¯χ),

    where τ(χ)=G(χ,1).

    Lemma 2.2. Let p be a prime and let χ be any Dirichlet character mod p. For any integers r,s, we have

    |p1a=1χ(a)e(ra+s¯ap)|2p.

    Proof. See Lemma 1 of [10].

    Lemma 2.3. Let p be an odd prime and an integer h with (h,p)=1, and let χ be any Dirichlet character mod p, then we have

    χmodpχχ1(1)=(1)m¯χ(h)(r=1G(χχ1,r)rm)(s=1G(χ,s)sn)p32ln2p.

    Proof. For any fixed parameter Np, according to Abel's identity, we have

    r=1G(χχ1,r)rm=1rNG(χχ1,r)rm+m+NN<ryG(χχ1,r)rm+1dy.

    Since |G(χ,r)|p12, we have

    1rNG(χχ1,r)rmp121rN1rmp12lnN,

    and from the estimates for trigonometric sums we have

    N<ryG(χχ1,r)=p1a=1χχ1(a)N<rye(arp)=p1a=1χχ1(a)e((N+1)ap)e(([y]+1)ap)1e(ap)p1a=11|sinπap|p1a=1paplnp,

    it follows that

    m+NN<ryG(χχ1,r)rm+1dyplnpNm,

    then we have

    χmodpχχ1(1)=(1)m¯χ(h)(+r=1G(χχ1,r)rm)(+s=1G(χ,s)sn)=χmodpχχ1(1)=(1)m¯χ(h)(1rNG(χχ1,r)rm+m+NN<ryG(χχ1,r)rm+1dy)×(1rNG(χ,s)sn+n+NN<ryG(χ,s)sn+1dy)=χmodpχχ1(1)=(1)m¯χ(h)(Nr=1G(χχ1,r)rm)(Ns=1G(χ,s)sn)+nχmodpχχ1(1)=(1)m¯χ(h)(Nr=1G(χχ1,r)rm)(+NN<ryG(χ,s)sn+1dy)+mχmodpχχ1(1)=(1)m¯χ(h)(Ns=1G(χ,s)sn)(+NN<ryG(χχ1,r)rm+1dy)+mn(+NNryG(χχ1,r)rm+1dy)(+NN<ryG(χ,s)sn+1dy)χmodpχχ1(1)=(1)m¯χ1(h)(Nr=1G(χχ1,r)rm)(Ns=1G(χ,s)sn)+O(p32lnNlnpmin(Nm,Nn)).

    Note that

    χmodpχχ1(1)=(1)mχ(a)={12ϕ(p),a1modp;(1)mχ1(1)12ϕ(p),a1modp;0,otherwise. (2.1)

    Then, from the orthogonality of Dirichlet characters we find

    χmodpχχ1(1)=(1)m¯χ1(h)(Nr=1G(χχ1,r)rm)(Ns=1G(χ,s)sn)=χmodpχχ1(1)=(1)m¯χ(h)(1rNp1a=1χχ1(a)e(rap)rm)(1sNp1b=1χ(b)e(sbp)sn)=1rN1sN1rmsnp1a=1p1b=1χ1(a)e(ra+sbp)χmodpχχ1(1)=(1)mχ(a)χ(b)¯χ(h)=ϕ(p)21rN1sN1rmsnp1a=1p1b=1abhmodpχ1(a)e(ra+sbp)+(1)mχ1(1)ϕ(p)21rN1sN1rmsnp1a=1p1b=1abhmodpχ1(a)e(ra+sbp).

    According to Lemma 2.2, we obtain

    χmodpχχ1(1)=(1)m¯χ(h)(Nr=1G(χχ1,r)rm)(Ns=1G(χ,s)sn)p32ln2N.

    Taking N=p2, we can get Lemma 2.3

    χmodpχχ1(1)=(1)m¯χ(h)(+r=1G(χχ1,r)rm)(+s=1G(χ,s)sn)p32ln2p.

    Lemma 2.4. Let p be a prime and let χ be any Dirichlet character mod p. For any integers m,n, we have

    χmodpχχ1(1)=(1)n|L(n,χχ1)|2|L(m,χ)|2=(p1)ζ(2m)ζ(2n)2ζ(2m+2n)|L(n+m,χ1)|2+O(p1min(n,m)exp(4lnplnlnp)).

    Proof. First, we assume that n>m, according to Abel's identity we can write

    L(n,χχ1)L(m,χ)=t=1χ(t)D(t)tm=Nt=1χ(t)D(t)tm+mNB(y,χ)ym+1dy,

    where D(t)=d|tχ1(d)dnm, B(y,χ)=N<tyχ(t)D(t). We know that

    B(y,χ)=N<tyχ(t)D(t)=N<tyχ(t)dtχ1(d)dnm=tyχ(t)ly/tχχ1(l)lnm+lyχχ1(l)lnmty/lχ(t)tNχ(t)lN/tχχ1(l)lnmlNχχ1(l)lnmtN/lχ(t)tyχ(t)lyχχ1(l)lnm+lNχχ1(l)lnmtNχ(t).

    It follows that

    |B(y,χ)|ylny,
    χmodpχχ1(1)=(1)n|B(y,χ)|2yϕ(p)ln2y.

    From the Cauchy inequality we have

    χmodpχχ1(1)=(1)n|mNB(y,χ)ym+1dy|2{mN1ym+1(χmodpχχ1(1)=(1)n|B(y,χ)|2)1/2dt}2{mNym12ϕ12(p)lnydt}2ϕ(p)ln2NN2m1.

    From (2.1), we have

    χmodpχχ1(1)=(1)n|Nn1=1χ(n1)D(n1)nm1|2=ϕ(p)21n1,n2Nn1n2(modp)(n1n2,p)=1D(n1)¯D(n2)nm1nm2+χ1(1)(1)nϕ(p)21n1,n2Nn1n2(modp)(n1n2,p)=1D(n1)¯D(n2)nm1nm2=ϕ(p)21n1N(n1,p)=1|D(n1)|2n2m1+O(ϕ(p)Nn2=1[N/p]l=1d(n2)d(lp+n2)nm2(lp+n2)m)+O(ϕ(p)p1n1=1d(n1)d(qn1)nm1(pn1)m)+O(ϕ(p)Nn1=1[N/p]l=1+n1/pd(n1)d(lpn1)nm1(lpn1)m)=ϕ(p)21n1(n1,p)=1|D(n1)|2n2m1+O(ϕ(p)pmexp(2lnNlnlnN)),

    noting that |D(n1)|d(n1)=tn11exp((1+ϵ)ln2lnNlnlnN), it follows that

    χmodpχχ1(1)=(1)n(Nt=1χ(t)D(t)tm)(mNB(y,χ)ym+1dy)(lnN)2N1ym+1(χmodpχ(1)=(1)n|B(y,χ)|)dyϕ(p)N12m(lnN)3.

    Taking N=p2 and ϕ(p)=p1, then

    χmodpχχ1(1)=(1)n|L(n,χχ1)|2|L(m,χ)|2=p121n1(n1,p)=1|D(n1)|2n2m1+O(p1mexp(4lnplnlnp)).

    From the Euler product formula, we can get

    n1=1(n1,p)=1|D(n1)|2n2m1=p1p(1+|D(p1)|2p2m1+|D(p21)|2p4m1++|D(pk1)|2p2mk1+)

    and

    D(pk1)=1+χ1(p1)pnm1+(χ1(p1)pnm1)2++(χ1(p1)pnm1)k=1(χ1(p1)pnm1)k+11χ1(p1)pnm1,

    and it is straightforward to show that

    n1=1(n1,p)=1|D(n1)|2n2m1=p1p111p2m1+1p2n2m111p2n1χ1(p1)pnm11χ1(p1)pn+m1¯χ1(p1)pnm11¯χ1(p1)pn+m1|1χ1(p1)pnm1|2=p1p11p2m+2n1(11p2m1)(11p2n1)1|1χ1(p1)pn+m1|2=ζ(2m)ζ(2n)ζ(2m+2n)|L(n+m,χ1)|2(11p2m)(11p2n)11p2m+2n=ζ(2m)ζ(2n)ζ(2m+2n)|L(n+m,χ1)|2+O(p2m).

    Similarly, we also have Lemma 2.4 for nm.

    Now, we prove our theorems by using the above lemmas.

    Proof of Theorem 1.1. Combining Lemma 2.1 and Lemma 2.3, we have

    C(h,m,n,p,χ)m!n!(2π)(m+n)p12ln2p.

    Proof of Theorem 1.2. From Lemma 2.1 and Lemma 2.4, we have

    p1h=1|C(h,m,n,p,χ)|2=16(m!n!)2(2πi)2(m+n)1ϕ2(p)p1h=1|χ1modpχχ1(1)=(1)m¯χ1(h)τ(χ1)τ(χχ1)L(m,¯χχ1)L(n,¯χ1)|2=16(m!n!)2(2πi)2(m+n)p2ϕ(p)χ1modpχχ1(1)=(1)m|L(m,¯χχ1)L(n,¯χ1)|2=8(m!n!)2p2(2πi)2(m+n)ζ(2m)ζ(2n)ζ(2m+2n)|L(m+n,χ)|2+O(p2min(n,m)exp(4lnplnlnp)).

    This proves Theorem 1.2.

    Proof of Theorem 1.4. From the definition of C(h,m,n,p,χ) and the orthogonality of Dirichlet characters, we have

    |C(h,m,n,p,χ)|2=|p1a=1χ(a)¯Bm(¯ap)¯Bn(ahp)|2={p1a=1χ(a)¯Bm(¯ap)¯Bn(ahp)}{p1b=1¯χ(b)¯Bm(¯bp)¯Bn(bhp)}=p1a=1p1b=1χ(a)¯Bm(¯abp)¯Bn(abhp)¯Bm(¯bp)¯Bn(bhp)=1ϕ(p)χ1modpp1a=1χ(a){p1b=1χ1(b)¯Bm(¯abp)¯Bm(bp)}×{p1c=1χ1(c)¯Bn(hcp)¯Bn(ahcp)}=1ϕ(p)χ1modp¯χ1(h)p1a=1χ(a){p1b=1χ1(b)¯Bm(¯abp)¯Bm(bp)}×{p1c=1χ1(c)¯Bn(cp)¯Bn(acp)},

    it follows that

    p1h=1|C(h,m,n,p,χ)|2=1ϕ(p)p1h=1χ1modp¯χ1(h)p1a=1χ(a){p1b=1χ1(b)¯Bm(¯abp)¯Bm(bp)}×{p1c=1χ1(c)¯Bn(cp)¯Bn(acp)}=p1a=1χ(a){p1b=1¯Bm(¯abp)¯Bm(bp)}{p1c=1¯Bn(cp)¯Bn(acp)}.

    Thus, from Theorem 1.2, we have

    p1h=1χ(h)S(¯h,m,m,p)S(h,n,n,p)=8(m!n!)2p2(2πi)2(m+n)ζ(2m)ζ(2n)ζ(2m+2n)|L(m+n,χ)|2+O(p2min(n,m)exp(4lnplnlnp)).

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    This work is supported by the National Natural Science Foundation of China (11971381).

    The authors declare that there are no conflicts of interest regarding the publication of this paper.



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