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Research article

The Minkowski’s inequality by means of a generalized fractional integral

  • We use the definition of a fractional integral, recently proposed by Katugampola, to establish a generalization of the reverse Minkowski's inequality. We show two new theorems associated with this inequality, as well as state and show other inequalities related to this fractional operator.

    Citation: J. Vanterler da C. Sousa, E. Capelas de Oliveira. The Minkowski’s inequality by means of a generalized fractional integral[J]. AIMS Mathematics, 2018, 3(1): 131-147. doi: 10.3934/Math.2018.1.131

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  • We use the definition of a fractional integral, recently proposed by Katugampola, to establish a generalization of the reverse Minkowski's inequality. We show two new theorems associated with this inequality, as well as state and show other inequalities related to this fractional operator.



    More and more fields of research have used fractional calculus to develop and find new applications. Similarly, q-calculus is involved in several engineering domains, physics, and in mathematics. The combination of fractional and q-calculus in geometric functions theory and some interesting applications were obtained by Srivastava [1].

    Jackson [2,3] established the q-derivative and the q-integral in the field of mathematical analysis via quantum calculus. The foundations of quantum calculus in the theory of geometric functions were laid by Srivastava [4]. Continued research in this field has led to the obtaining of numerous q-analogue operators, such as the q-analogue of the Sălăgean differential operator [5], giving new applications in [6,7,8]; the q-analogue of the Ruscheweyh differential operator introduced by Răducanu and Kanas [9] and studied by Mohammed and Darus [10] and Mahmood and Sokół [11]; and the q -analogue of the multiplier transformation [12,13].

    This study involves an operator defined by applying the Riemann-Liouville fractional integral to the q-analogue of the multiplier transformation. Many operators have been defined and studied in recent years by using the Riemann-Liouville or Atagana-Baleanu fractional integrals.

    First, we recall the classically used notations and notions from geometric functions theory.

    Working on the open unit disc U={zC:|z|<1}, we establish here the class of analytic functions denoted by H(U) and its subclasses H[a,n] containing the functions fH(U) defined by f(z)=a+anzn+an+1zn+1+, with zU, aC, nN, as well as An, containing the functions fH(U) of the form f(z)=z+an+1zn+1+, zU. When n=1, the notation A1=A is used.

    We also recall the Riemann-Liouville fractional integral definition introduced in [13,15]:

    Definition 1. ([13,15]) The fractional integral of order λ applied to the analytic function f in a simply-connected region of the z -plane which contains the origin is defined by

    Dλzf(z)=1Γ(λ)z0f(t)(zt)1λdt,

    where λ>0 and the multiplicity of (zt)λ1 is removed by the condition that log(zt) is real when (zt)>0.

    The q-analogue of the multiplier transformation is defined below.

    Definition 2 ([13]) The q-analogue of the multiplier transformation, denoted by Im,lq, has the following form:

    Im,lqf(z)=z+j=2([l+j]q[l+1]q)majzj,

    where q(0,1), m,lR, l>1, and f(z)=z+j=2ajzjA, zU.

    Remark 1. We notice that limq1Im,lqf(z)=limq1(z+j=2([l+j]q[l+1]q)majzj)=z+j=2(l+jl+1)majzj =I(m,1,l). The operator I(m,1,l) was studied by Cho and Srivastava [16] and Cho and Kim [17]. The operator I(m,1,1) was studied by Uralegaddi and Somanatha [18], and the operator I(α,λ,0) was introduced by Acu and Owa [19]. Cătaş [20] studied the operator Ip(m,λ,l) which generalizes the operator I(m,λ,l). Alb Lupaş studied the operator I(m,λ,l) in [21,22,23].

    Now, we introduce definitions from the differential subordination and differential superordination theories.

    Definition 3. ([24]) Between the analytic functions f and g there is a differential subordination, denoted f(z)g(z), if there exists ω, a Schwarz analytic function with the properties |ω(z)|<1, zU and ω(0)=0, such that f(z)=g(ω(z)), zU. In the special case where g is an univalent function in U, the above differential subordination is equivalent to f(U)g(U) and f(0)=g(0).

    Definition 4. ([24]) Considering a univalent function h in U and ψ:C3×UC, when the analytic function p satisfies the differential subordination

    ψ(p(z),zp(z),z2p(z);z)h(z),  zU, (1.1)

    then p is a solution of the differential subordination. When pg for all solutions p, the univalent function g is a dominant of the solutions. A dominant ˜g with the property ˜gg for every dominant g is called the best dominant of the differential subordination.

    Definition 5. ([25]) Considering an analytic function h in U and φ:C3ׯUC, when p and φ(p(z),zp(z),z2p(z);z) are univalent functions in U fulfilling the differential superordination

    h(z)φ(p(z),zp(z),z2p(z);z),    (1.2)

    then p is a solution of the differential superordination. When gp for all solutions p, the analytic function g is a subordinant of the solutions. A subordinant ˜g with the property g˜g for every subordinant g is called the best subordinant of the differential superordination.

    Definition 6. ([24]) Q denotes the class of injective functions f analytic on ¯UE(f), with the property f(ζ)0 for ζUE(f), when E(f)={ζU:limzζf(z)=}.

    The obtained results from this paper are constructed based on the following lemmas.

    Lemma 1. ([24]) Considering the univalent function g in U and the analytic functions θ, η in a domain Dg(U), such that η(w)0, wg(U), define the functions G(z)=zg(z)η(g(z)) and h(z)=θ(g(z))+G(z). Assuming that G is starlike univalent in U and Re(zh(z)G(z))>0, zU, when the analytic function p having the properties p(U)D and p(0)=g(0), satisfies the differential subordination θ(p(z))+zp(z)η(p(z))θ(g(z))+zg(z)η(g(z)), for zU, then pg and g is the best dominant.

    Lemma 2. ([26]) Considering the convex univalent function g in U and the analytic functions θ, η in a domain Dg(U), define the function G(z)=zg(z)η(g(z)). Assuming that G is starlike univalent in U and Re(θ(g(z))η(g(z)))>0, zU, when pH[g(0),1]Q, with p(U)D, the function θ(p(z))+zp(z)η(p(z)) is univalent in U, and the differential superordination θ(g(z))+zg(z)η(g(z))θ(p(z))+zp(z)η(p(z)) is satisfied, then gp and g is the best subordinant.

    The operator obtained by applying the the Riemann-Liouville fractional integral to the q-analogue of the multiplier transformation is written as follows:

    Definition 7. Let q,m,l be real numbers, q(0,1), l>1, and λN. The fractional integral applied to the q-analogue of the multiplier transformation is defined by

    DλzIm,lqf(z)=1Γ(λ)z0Im,lqf(t)(zt)1λdt= (2.1)
    1Γ(λ)z0t(zt)1λdt+j=2([l+j]q[l+1]q)majz0tj(zt)1λdt.

    After a laborious computation, we discover that the fractional integral applied to the q-analogue of the multiplier transformation takes the following form:

    DλzIm,lqf(z)=1Γ(λ+2)zλ+1+j=2([l+j]q[l+1]q)mΓ(j+1)Γ(j+λ+1)ajzj+λ, (2.2)

    when f(z)=z+j=2ajzjA. We note that DλzIm,lqf(z)H[0,λ+1].

    Remark 2. When q1, we obtain the classical case, and the fractional integral applied to the multiplier transformation is defined by

    DλzI(m,1,l)f(z)=1Γ(λ)z0I(m,1,l)f(t)(zt)1λdt= (2.3)
    1Γ(λ)z0t(zt)1λdt+j=2(l+jl+1)majz0tj(zt)1λdt,

    which, after several calculus can be written in the form

    DλzI(m,1,l)f(z)=1Γ(λ+2)zλ+1+j=2(l+jl+1)mΓ(j+1)Γ(j+λ+1)ajzj+λ, (2.4)

    when f(z)=z+j=2ajzjA. We note that DλzI(m,1,l)f(z)H[0,λ+1].

    The main subordination result product regarding the operator introduced in Definition 7 is exposed in the following theorem:

    Theorem 1. Consider fA and g an analytic function univalent in U with the property that g(z)0, zU, with real numbers q,m,l, q(0,1), l>1, and λ,nN. Assuming that zg(z)g(z) is a starlike function univalent in U and

    Re(1+bdg(z)+2cd(g(z))2zg(z)g(z)+zg(z)g(z))>0, (2.5)

    for a,b,c,dC, d0, zU, denote

    ψm,l,qλ(n,a,b,c,d;z):=a+b[DλzIm,lqf(z)z]n+ (2.6)
    c[DλzIm,lqf(z)z]2n+dn[z(DλzIm,lqf(z))DλzIm,lqf(z)1].

    If the differential subordination

    ψm,l,qλ(n,a,b,c,d;z)a+bg(z)+c(g(z))2+dzg(z)g(z), (2.7)

    is satisfied by the function g, for a,b,c,dC, d0, then the differential subordination

    (DλzIm,lqf(z)z)ng(z), (2.8)

    holds and g is the best dominant for it.

    Proof. Setting p(z):=(DλzIm,lqf(z)z)n, zU, z0, and differentiating with respect to z, we get

    p(z)=n(DλzIm,lqf(z)z)n1[(DλzIm,lqf(z))zDλzIm,lqf(z)z2]=
    n(DλzIm,lqf(z)z)n1(DλzIm,lqf(z))znzp(z)

    and

    zp(z)p(z)=n[z(DλzIm,lqf(z))DλzIm,lqf(z)1].

    Defining the functions θ and η by θ(w):=a+bw+cw2 and η(w):=dw, it can be easily certified that θ is analytic in C, η is analytic in C{0}, and that η(w)0, wC{0}.

    Considering the functions G(z)=zg(z)η(g(z))=dzg(z)g(z) and

    h(z)=θ(g(z))+G(z)=a+bg(z)+c(g(z))2+dzg(z)g(z),

    we deduce that G(z) is starlike univalent in U.

    Differentiating the function h with respect to z we get

    h(z)=bg(z)+2cg(z)g(z)+d(g(z)+zg(z))g(z)z(g(z))2(g(z))2

    and

    zh(z)G(z)=zh(z)dzg(z)g(z)=1+bdg(z)+2cd(g(z))2zg(z)g(z)+zg(z)g(z).

    The condition

    Re(zh(z)G(z))=Re(1+bdg(z)+2cd(g(z))2zg(z)g(z)+zg(z)g(z))>0

    is satisfied by relation (2.5), and we deduce that

    a+bp(z)+c(p(z))2+dzp(z)p(z)=a+b[DλzIm,lqf(z)z]n+
    c[DλzIm,lqf(z)z]2n+dγ[z(DλzIm,lqf(z))DλzIm,lqf(z)1]=ψm,l,qλ(n,a,b,c,d;z),

    which is the function from relation (2.6).

    Rewriting the differential subordination (2.7), we obtain

    a+bp(z)+c(p(z))2+dzp(z)p(z)a+bg(z)+c(g(z))2+dzg(z)g(z).

    The hypothesis of Lemma 1 being fulfilled, we get the conclusion p(z)g(z), written as

    (DλzIm,lqf(z)z)ng(z)

    and g is the best dominant.

    Corollary 1. Suppose that the relation (2.5) is fulfilled for real numbers q,m,l, q(0,1), l>1, and λ,nN. If the differential subordination

    ψm,l,qλ(n,a,b,c,d;z)a+bαz+1βz+1+c(αz+1βz+1)2+d(αβ)z(αz+1)(βz+1)

    is verified for a,b,c,dC, d0, 1β<α1, and the function ψm,l,qλ is given by relation (2.6), then the differential subordination

    (DλzIm,lqf(z)z)nαz+1βz+1

    is satisfied with the function g(z)=αz+1βz+1 as the best dominant.

    Proof. Considering in Theorem 1 the function g(z)=αz+1βz+1, with 1β<α1, the corollary is verified.

    Corollary 2. Assume that relation (2.5) is satisfied for real numbers q,m,l, q(0,1), l>1, and λ,nN. If the differential subordination

    ψm,l,qλ(n,a,b,c,d;z)a+b(z+11z)s+c(z+11z)2s+2sdz1z2

    holds for a,b,c,dC, 0<s1, d0, and the function ψm,l,qλ is defined by relation (2.6), then the differential subordination

    (DλzIm,lqf(z)z)n(z+11z)s

    is satisfied with the function g(z)=(z+11z)s as the best dominant.{}

    Proof. Considering in Theorem 1 the function g(z)=(z+11z)s, with 0<s1, the corollary is obtained.

    When q1 in Theorem 1, we get the classical case:

    Theorem 2. Consider fA and g an analytic function univalent in U with the property that g(z)0, zU, with real numbers m,l, l>1, and λ,nN. Assuming that zg(z)g(z) is a starlike function univalent in U and

    Re(1+bdg(z)+2cd(g(z))2zg(z)g(z)+zg(z)g(z))>0, (2.9)

    for a,b,c,dC, d0, zU, denote

    ψm,l,λ(n,a,b,c,d;z):=a+b[DλzI(m,1,l)f(z)z]n+ (2.10)
    c[DλzI(m,1,l)f(z)z]2n+dn[z(DλzI(m,1,l)f(z))DλzI(m,1,l)f(z)1].

    If the differential subordination

    ψm,l,λ(n,a,b,c,d;z)a+bg(z)+c(g(z))2+dzg(z)g(z), (2.11)

    is satisfied by the function g, for a,b,c,dC, d0, then the differential subordination

    (DλzI(m,1,l)f(z)z)ng(z) (2.12)

    holds and g is the best dominant for it.

    Proof. The proof of the theorem follows the same steps as the proof of Theorem 1 and it is omitted.

    The corresponding superordination results regarding the operator introduced in Definition 7 are exposed in the following:

    Theorem 3. Consider fA and g an analytic function univalent in U with the properties g(z)0 and zg(z)g(z) is starlike univalent in U, with real numbers q,m,l, q(0,1), l>1, and λ,nN. Assuming that

    Re(2cd(g(z))2+bdg(z))>0,forb,c,dC,d0 (2.13)

    and the function ψm,l,qλ(n,a,b,c,d;z) is defined in relation (2.6), if the differential superordination

    a+bg(z)+c(g(z))2+dzg(z)g(z)ψm,l,qλ(n,a,b,c,d;z) (2.14)

    is fulfilled for the function g, for a,b,c,dC, d0, then the differential superordination

    g(z)(DλzIm,lqf(z)z)n (2.15)

    holds and g is the best subordinant for it.

    Proof. Set p(z):=(DλzIm,lqf(z)z)n, zU, z0.

    Defining the functions θ and η by θ(w):=a+bw+cw2 and η(w):=dw, it is evident that η(w)0, wC{0} and we can certify that θ is analytic in C and η is analytic in C{0}.

    With easy computation, we get that

    θ(g(z))η(g(z))=g(z)[b+2cg(z)]g(z)d

    and relation (2.13) can be written as

    Re(θ(g(z))η(g(z)))=Re(2cd(g(z))2+bdg(z))>0,

    for b,c,dC, d0.

    Following the same computations as in the proof of Theorem 1, the differential superordination (2.14) can be written as

    a+bg(z)+c(g(z))2+dzg(z)g(z)a+bp(z)+c(p(z))2+dzp(z)p(z).

    The hypothesis of Lemma 2 being fulfilled, we obtain the conclusion

    g(z)p(z)=(DλzIm,lqf(z)z)n

    and g is the best subordinant.

    Corollary 3. Assume that relation (2.13) is verified for real numbers q,m,l, q(0,1), l>1, and λ,nN. If the differential superordination

    a+bαz+1βz+1+c(αz+1βz+1)2+d(αβ)z(αz+1)(βz+1)ψm,l,qλ(n,a,b,c,d;z)

    is satisfied for a,b,c,dC, d0, 1β<α1, and the function ψm,l,qλ is defined by the relation (2.6), then the differential superordination

    αz+1βz+1(DλzIm,lqf(z)z)n

    holds with the function g(z)=αz+1βz+1 as the best subordinant.

    Proof. Considering in Theorem 3 the function g(z)=αz+1βz+1, with 1β<α1, the corollary is proved.

    Corollary 4. Suppose that relation (2.13) is fulfilled for real numbers q,m,l, q(0,1), l>1, and λ,nN. If the differential superordination

    a+b(z+11z)s+c(z+11z)2s+2sdz1z2ψm,l,qλ(n,a,b,c,d;z)

    is satisfied for a,b,c,dC, 0<s1, d0, and the function ψm,l,qλ is given by the relation (2.6), then the differential superordination

    (z+11z)s(DλzIm,lqf(z)z)n

    is satisfied with the function g(z)=(z+11z)s as the best subordinant.

    Proof. Considering in Theorem 3 the function g(z)=(z+11z)s, with 0<s1, the corollary is obtained.

    When q1 in Theorem 3, we get the classical case:

    Theorem 4. Consider fA and g an analytic function univalent in U with the properties g(z)0 and zg(z)g(z) is starlike univalent in U, with real numbers m,l, l>1, and λ,nN. Assuming that

    Re(2cd(g(z))2+bdg(z))>0,forb,c,dC,d0, (2.16)

    and the function ψm,l,λ(n,a,b,c,d;z) is defined in relation (2.10), if the differential superordination

    a+bg(z)+c(g(z))2+dzg(z)g(z)ψm,l,λ(n,a,b,c,d;z) (2.17)

    is fulfilled for the function g, for a,b,c,dC, d0, then the differential superordination

    g(z)(DλzI(m,1,l)f(z)z)n (2.18)

    holds and g is the best subordinant for it.

    Proof. The proof of the theorem follows the same steps as the proof of Theorem 3 and it is omitted.

    The sandwich-type result is obtained by combining Theorems 1 and 3.

    Theorem 5. Consider fA and g1, g2 analytic functions univalent in U with the properties that g1(z)0, g2(z)0, zU, and, respectively, zg1(z)g1(z), zg2(z)g2(z) are starlike univalent, with real numbers q,m,l, q(0,1), l>1, and λ,nN. Assuming that relation (2.5) is verified by the function g1 and the relation (2.13) is verified by the function g2, and the function ψm,l,qλ(n,a,b,c,d;z) defined by relation (2.6) is univalent in U, if the sandwich-type relation

    a+bg1(z)+c(g1(z))2+dzg1(z)g1(z)ψm,l,qλ(n,a,b,c,d;z)a+bg2(z)+c(g2(z))2+dzg2(z)g2(z)

    is satisfied for a,b,c,dC, d0, then the below sandwich-type relation

    g1(z)(DλzIm,lqf(z)z)ng2(z)

    holds for g1 as the best subordinant and g2 the best dominant.

    Considering in Theorem 5 the functions g1(z)=α1z+1β1z+1, g2(z)=α2z+1β2z+1, with 1β2<β1<α1<α21, the following corollary holds.

    Corollary 5. Suppose that relations (2.5) and (2.13) are fulfilled for real numbers q,m,l, q(0,1), l>1, and λ,nN. If the sandwich-type relation

    a+bα1z+1β1z+1+c(α1z+1β1z+1)2+d(α1β1)z(α1z+1)(β1z+1)ψm,l,qλ(n,a,b,c,d;z)
    a+bα2z+1β2z+1+cχ(α2z+1β2z+1)2+d(α2β2)z(α2z+1)(β2z+1)

    is satisfied for a,b,c,dC, d0, 1β2β1<α1α21, and the function ψm,l,qλ is defined by the relation (2.6), then the following sandwich-type relation

    α1z+1β1z+1(DλzIm,lqf(z)z)nα2z+1β2z+1

    holds for g1(z)=α1z+1β1z+1 as the best subordinant and g2(z)=α2z+1β2z+1 the best dominant.

    Considering in Theorem 5 the functions g1(z)=(z+11z)s1, g2(z)=(z+11z)s2, with 0<s1,s21, the following corollary holds.

    Corollary 6. Assume that the relations (2.5) and (2.13) are satisfied for real numbers q,m,l, q(0,1), l>1, and λ,nN. If the sandwich-type relation

    a+b(z+11z)s1+c(z+11z)2s1+2s1dz1z2ψm,l,qλ(n,a,b,c,d;z)
    a+b(z+11z)s2+v(z+11z)2s2+2s2dz1z2

    holds for a,b,c,dC, d0, 1β2β1<α1α21, and the function ψm,l,qλ is defined by the relation (2.6), then the following sandwich-type relation

    (z+11z)s1(DλzIm,lqf(z)z)n(z+11z)s2

    is satisfied for g1(z)=(z+11z)s1 as the best subordinant and g2(z)=(z+11z)s2 the best dominant.

    The sandwich-type result is obtained by combining Theorems 2 and 4 for the classical case when q1.

    Theorem 6. Consider fA and g1, g2 analytic functions univalent in U with the properties that g1(z)0, g2(z)0, zU, and, respectively, zg1(z)g1(z), zg2(z)g2(z) are starlike univalent, with real numbers m,l, l>1, and λ,nN. Assuming that relation (2.9) is verified by the function g1 and relation (2.16) is verified by the function g2, and the function ψm,l,λ(n,a,b,c,d;z) from relation (2.10) is univalent in U, if the sandwich-type relation

    a+bg1(z)+c(g1(z))2+dzg1(z)g1(z)ψm,l,λ(n,a,b,c,d;z)a+bg2(z)+c(g2(z))2+dzg2(z)g2(z),

    is satified for a,b,c,dC, d0, then the below sandwich-type relation

    g1(z)(DλzI(m,1,l)f(z)z)ng2(z)

    holds for g1 as the best subordinant and g2 the best dominant.

    The results presented in this paper are determined as applications of fractional calculus combined with q-calculus in geometric functions theory. We obtain a new operator described in Definition 7 by applying a fractional integral to the q-analogue of the multiplier transformation. The new fractional q-analogue of the multiplier transformation operator introduced in this paper yields new subordination and superordination results.

    The subordination theory used in Theorem 1 gives the best dominant of the differential subordination and, considering well-known functions in geometric functions theory as the best dominant, some illustrative corollaries are obtained. Using the duality, the superordination theory used in Theorem 3 gives the best subordinant of the differential superordination, and illustrative corollaries are established taking the same well-known functions. Combining Theorem 1 and Theorem 3, we present a sandwich-type theorem involving the two dual theories of differential subordination and superordination. Considering the functions studied in the previous corollaries, we establish the other sandwich-type results. The classical case when q1 is also presented.

    For future studies, using the fractional integral of the q-analogue of the multiplier transformation introduced in this paper, and following [27] and [28], we can define q- subclasses of univalent functions and study some properties, such as coefficient estimates, closure theorems, distortion theorems, neighborhoods, radii of starlikeness, convexity, and close-to-convexity of functions belonging to the defined subclass.

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

    The publication of this research was supported by the University of Oradea, Romania and Mustansiriyah University (www.uomustansiriyah.edu.iq) in Baghdad, Iraq. All thanks to their support.

    The authors declare that they have no conflicts of interest.

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