The object of the present paper is to obtain several fuzzy differential subordinations associated with Linear operator analytic functions for which we give some properties.
Mathematics Subject Classification (2010): 30C45, 30A20.
Received 10 April 2020; Revised 16 July 2020. Published Online: 2023-03-20. Published Print: 2023-04-3

ALB LUPAȘ, Alina

EL-DEEB, Sheza M.

English

Studia Universitatis Babeş-Bolyai

Stud. Univ. Babeş-Bolyai Math. 68(2023), No. 1, 151–160DOI: 10.24193/subbmath.2023.1.11Fuzzy differential subordinations connectedwith convolutionSheza M. El-Deeb and Alina Alb LupaşAbstract. The object of the present paper is to obtain several fuzzy differentialsubordinations associated with Linear operatorDmn,δ,gf(z) = z +∞∑j=2[1 + (j − 1) cn(δ)]m ajbjzj .Using the operator Dmn,δ,g, we also introduce a class HFn,m,δ (η, g) of univalentanalytic functions for which we give some properties.Mathematics Subject Classification (2010): 30C45, 30A20.Keywords: Fuzzy differential subordination, fuzzy best dominant, binomial series,linear differential operator, convolution.1. IntroductionLet Ω⊂ C, H(Ω) the class of holomorphic functions on Ω and denote by Hd(Ω)the class of holomorphic and univalent functions on Ω. In this paper, we denote byH(∆) the class of holomorphic functions in the open unit disk ∆ = {z ∈ C : |z| < 1}with B∆ = {z ∈ C : |z| = 1} the boundary of the unit disk. For β ∈ C and d ∈ N, wedenoteH [β, d] =f ∈ H(∆) : f(z) = β +∞∑j=d+1ajzj , z ∈ ∆ ,Ad =f ∈ H(∆) : f(z) = z +∞∑j=d+1ajzj , z ∈ ∆ with A1 = A,andS = {f ∈ A : f is a univalent function in ∆} .Received 10 April 2020; Revised 16 July 2020.152 Sheza M. El-Deeb and Alina Alb LupaşWe denote byC ={f ∈ A : <(1 +zf′′(z)f ′(z))> 0, z ∈ ∆},the set of convex functions in ∆.Definition 1.1. [4, 11] Let f1 and f2 are analytic function in ∆, then f1 is subordinateto f2, written f1 ≺ f2 if there exists a Schwarz function w, which is analytic in ∆with w(0) = 0 and |w(z)| < 1 for all z ∈ ∆, such that f1(z) = f2(w(z)). Furthermore,if the function f2 is univalent in ∆, then we have the following equivalence:f1(z) ≺ f2(z)⇔ f1(0) = f2(0) and f1(∆) ⊂ f2(∆).In order to introduce the notion of fuzzy differential subordination, we use thefollowing definitions and propositions:Definition 1.2. [10] Fuzzy subset of Y is a pair (B,FB), with FB : Y → [0, 1] andB = {x ∈ Y : 0 < FB(x) ≤ 1}. (1.1)The support of the fuzzy set (B,FB) is the set B and the membership function of(B,FB) is FB.Proposition 1.3. [12] (i) If (B,FB) = (U ,FU ), then we have B = U , whereB = sup (B,FB) and U = sup (U ,FU ) ;(ii) If (B,FB) ⊆ (U ,FU ), then we have B ⊆ U , whereB = sup (B,FB) and U = sup (U ,FU ) .Let f, g ∈ H(Ω), we denote byf (Ω) ={f(z) : 0 < Ff(Ω)f(z) ≤ 1, z ∈ Ω}= sup(f (Ω) ,Ff(Ω)), (1.2)andg (Ω) ={g(z) : 0 < Fg(Ω)g(z) ≤ 1, z ∈ Ω}= sup(g (Ω) ,Fg(Ω)). (1.3)Definition 1.4. [12] Let z0 ∈ Ω be a fixed point and let the functions f, g ∈ H(Ω). Thefunction f is said to be fuzzy subordinate to g and write f ≺F g or f(z) ≺F g(z), ifare satisfied the following conditions:(i) f(z0) = g(z0)(ii) Ff(Ω)f(z) ≤ Fg(Ω)g(z), z ∈ Ω.Proposition 1.5. [12] Assume that z0 ∈ Ω is a fixed point and the functions f, g ∈H(Ω). If f(z) ≺F g(z), z ∈ Ω, then(i) f(z0) = g(z0)(ii) f (Ω) ⊆ g (Ω) , Ff(Ω)f(z) ≤ Fg(Ω)g(z), z ∈ Ω,where f (Ω) and g (Ω) are defined by (1.2) and (1.3), respectively.Definition 1.6. [13] Assume that Φ : C3 × ∆ → C and h ∈ S, with Φ (α, 0, 0; 0) =h(0) = α. If p is analytic in ∆, with p(0) = α and satisfies the second order fuzzydifferential subordinationFΦ(C3×∆)Φ(p(z), zp′(z), z2p′′(z); z)≤ Fh(∆)h(z),Fuzzy differential subordinations connected with convolution 153i.e. Φ(p(z), zp′(z), z2p′′(z); z)≺F h(z), z ∈ ∆, (1.4)then p is said to be a fuzzy solution of the fuzzy differential subordination. Theunivalent function q is called a fuzzy dominant of the fuzzy solutions for the fuzzydifferential subordination ifFp(∆)p(z) ≤ Fq(∆)q(z), i.e. p(z) ≺F q(z), z ∈ ∆for all p satisfying (1.4).A fuzzy dominant q̃ that satisfiesFq̃(∆)q̃(z) ≤ Fq(∆)q(z), i.e. q̃(z) ≺F q(z), z ∈ ∆for all fuzzy dominants q of (1.4) is called the fuzzy best dominant of (1.4).Making use the binomial series(1− δ)n =n∑i=0(ni)(−1)i δi (n ∈ N = {1, 2, ...}) ,for f ∈ A, we introduced the linear differential operator as follows:D0n,δ,gf(z) = (f ∗ g) (z),D1n,δ,gf(z) = Dn,δ,gf(z) = (1− δ)n(f ∗ g) (z) + [1− (1− δ)n] z (f ∗ g)′(z)= z +∞∑j=2[1 + (j − 1) cn(δ)] ajbjzj...Dmn,δ,gf(z) = Dn,δ,g(Dm−1n,δ,gf(z))= (1− δ)nDm−1n,δ,gf(z) + [1− (1− δ)n] z(Dm−1n,δ,gf(z))′= z +∞∑j=2[1 + (j − 1) cn(δ)]m ajbjzj (1.5)(δ > 0, n ∈ N, m ∈ N0 = N ∪ {0}) ,wherecn(δ) =n∑i=1(ni)(−1)i+1 δi (n ∈ N) .From (1.5), we obtain thatcn(δ) z(Dmn,δ,gf(z))′= Dm+1n,δ,gf(z)− [1− cn(δ)]Dmn,δ,gf(z).By specializing the parameters n, δ and bj , we note that(i) Putting bj = 1 (or g(z) =z1−z ), then Dmn,δ, z1−z= Dmn,δ defined by Yousef et al. [17].(ii) Putting bj = 1 (or g(z) =z1−z ) and n = 1, then Dm1,δ, z1−z= Dmδ defined byAl-Oboudi [3].154 Sheza M. El-Deeb and Alina Alb Lupaş(iii) Putting bj = 1 (or g(z) =z1−z ) and n = δ = 1, then Dm1,1, z1−z= Dm defined bySălăgean.[15].(iv) Putting bj =(`+1`+j)α(α > 0, ` > −1) and n = 1, then Dm1,δ,g = Im,α`,δ f(z) definedby El-Deeb and Lupaş [6].(v) Putting bj =(α+1α+j)nmj−1(j−1)!e−m (m,α ≥ 0, n ∈ N0) and m = 0, then D0n,δ,g =Hnα,mf(z) defined by El-Deeb and Oros [9].(vi) Putting bj =(−1)k−1Γ(υ+1)4k−1(k−1)!Γ(k+υ) ·[k,q]![λ+1,q]k−1, (υ > 0, λ > −1, 0 < q < 1) studied byEl-Deeb and Bulboacă [7] and El-Deeb [5], we obtain the operator Nm,λ,qυ,n,δ , defined asfollows:Nm,λ,qυ,n,δ f(z) = z +∞∑j=2[1 + (j − 1) cn(δ)]m (−1)j−1Γ(υ + 1)4j−1(j − 1)!Γ(j + υ)ajzj(λ > −1; 0 < q < 1; δ, υ > 0; n ∈ N; m ∈ N0) ;(vi) Putting bj =(`+1`+j)α· [k,q]![λ+1,q]k−1 , (α > 0, n ≥ 0, λ > −1, 0 < q < 1) studied byEl-Deeb and Bulboacă [8] and Srivastava and El-Deeb [16], we obtain the operatorMm,λ,q`,n,δ,α, defined as follows:Mm,λ,q`,n,δ,αf(z) = z +∞∑j=2[1 + (j − 1) cn(δ)]m(n+ 1n+ k)α[k, q]![λ+ 1, q]k−1ajzj(α > 0; λ > −1; ` ≥ 0; 0 < q < 1; δ > 0; n ∈ N; m ∈ N0) .2. PreliminaryTo prove our results, we need the following lemmas.Lemma 2.1. [11] Let ψ ∈ A andG(z) = 1zz∫0ψ(t)dt, z ∈ ∆.If <{1 + zψ′′(z)ψ′ (z)}> −12 , z ∈ ∆, then G ∈ K.Lemma 2.2. [14, Theorem 2.6] Let ψ be a convex function with ψ(0) = β and ν ∈C∗ = C\{0} with < (ν) ≥ 0. If p ∈ H [β, d] with p(0) = β, Φ : C2 ×∆→ C,Φ(p(z), zp′(z); z)= p(z) +1νzp′(z)is analytic function in ∆ andFΦ(C2×∆)(p(z) +1νzp′(z))≤ Fh(∆)h(z) → p(z) +1νzp′(z) ≺F h(z), z ∈ ∆,thenFp(∆)p(z) ≤ Fq(∆)q(z) ≤ Fh(∆)h(z) → p(z) ≺F q(z), z ∈ ∆,Fuzzy differential subordinations connected with convolution 155whereq(z) =νdzνdz∫0ψ(t)tνd−1dt, z ∈ ∆.The function q is convex and it is the fuzzy best dominant.Lemma 2.3. [14, Theorem 2.7] Let g be a convex function in ∆ andψ(z) = g(z) + dγzg′(z),where z ∈ ∆, d ∈ N and γ > 0. Ifp(z) = g(0) + pd zd + pd+1 zd+1 + ...belongs to H(∆), andFp(∆)(p(z) + γzp′(z))≤ Fψ(∆)ψ(z) → p(z) + γzp′(z) ≺F ψ(z), z ∈ ∆,thenFp(∆) (p(z)) ≤ Fg(∆)g(z) → p(z) ≺F g(z), z ∈ ∆.This result is sharp.For the general theory of fuzzy differential subordination and its applications,we refer the reader to [1, 2].In the next section, we obtain several fuzzy differential subordinations associatedwith the diferential operator Dmn,δ,gf(z) by using the method of fuzzy differentialsubordination.3. Main resultsAssume that η ∈ [0, 1), δ > 0, n ∈ N, m ∈ N0, λ > 0 and z ∈ ∆ are mentionedthrough this paper.By using the integral operator Dmn,δ,g, we define a class of analytic functions andwe derive several fuzzy differential subordinations for this class.Definition 3.1. Let the function f ∈ A belongs to the class HFn,m,δ (η, g) for allη ∈ [0, 1), n ∈ N0, m > 0 and α ≥ 0 if it satisfies the inequality:F(Dmn,δ,gf)′(∆)(Dmn,δ,gf(z))′> η, (z ∈ ∆) .Theorem 3.2. Let k belongs to C in ∆ and suppose that h(z) = k(z) + 1λ+2zk′(z).If f ∈ HFn,m,δ (η, g) andG(z) = Iλf(z) =λ+ 2zλ+1z∫0tλf(t)dt, (3.1)thenF(Dmn,δ,gf)′(∆)(Dmn,δ,gf(z))′≤ Fh(∆)h(z)→(Dmn,δ,gf(z))′≺F h(z), (3.2)156 Sheza M. El-Deeb and Alina Alb LupaşimpliesF(Dmn,δ,gG)′(∆)(Dmn,δ,gG(z))′≤ Fk(∆)k(z)→(Dmn,δ,gG(z))′≺F k(z),and this result is sharp.Proof. Sincezλ+1G(z) = (λ+ 2)z∫0tλf(t)dt,by differentiating, it obtain(λ+ 1)G(z) + zG′(z) = (λ+ 2) f(z),and(λ+ 1)Dmn,δ,gG(z) + z(Dmn,δ,gG(z))′= (λ+ 2)Dmn,δ,gf(z), (3.3)and also, by differentiating (3.3) we obtain(Dmn,δ,gG(z))′+1(λ+ 2)z(Dmn,δ,gG(z))′′=(Dmn,δ,gf(z))′(3.4)By using (3.4), the fuzzy differential subordination (3.2) isF(Dmn,δ,gf)′(∆)((Dmn,δ,gG(z))′+1(λ+ 2)z(Dmn,δ,gG(z))′′)≤ Fh(∆)(k(z) +1(λ+ 2)zk′(z)). (3.5)We denoteq(z) =(Dmn,δ,gG(z))′, so q ∈ H [1, n] . (3.6)Putting (3.6) in (3.5), we haveF(Dmn,δ,gf)′(∆)(q(z) +1(λ+ 2)zq′(z))≤ Fh(∆)(k(z) +1(λ+ 2)zk′(z)), (3.7)and applying Lemma (2.3), we haveFq(∆)q(z) ≤ Fk(∆)k(z), i.e F(Dmn,δ,gG(z))′(∆)(Dmn,δ,gG(z))′≤ Fk(∆)k(z),therefore(Dmn,δ,gG(z))′≺F k(z), and k is the fuzzy best dominant. Theorem 3.3. Assume that h(z) = 1+(2η−1)z1+z , η ∈ [0, 1) , λ > 0 and Iλ is given by(3.1), thenIλ[HFn,m,δ (η, g)]⊂ HFn,m,δ (η∗, g) , (3.8)whereη∗= 2η − 1 + (λ+ 2) (2− 2η)1∫0tλ+2t+ 1dt . (3.9)Fuzzy differential subordinations connected with convolution 157Proof. A function h belongs to C and using the same technique in the proof of Theorem3.2, we obtain from the hypothesis of Theorem 3.3 thatFq(∆)(q(z) +1(λ+ 2)zq′(z))≤ Fh(∆)h(z),where q(z) is defined in (3.6). By using Lemma 2.2, we obtainFq(∆)q(z) ≤ Fk(∆)k(z) ≤ Fh(∆)h(z),which impliesF(Dmn,δ,gG)′(∆)(Dmn,δ,gG(z))′≤ Fk(∆)k(z) ≤ Fh(∆)h(z),wherek(z) =λ+ 2zλ+2z∫0tλ+11 + (2η − 1) t1 + tdt= (2η − 1) + (λ+ 2) (2− 2η)zλ+2z∫0tλ+11 + tdt.k belongs to C and k (∆) is symmetric with respect to the real axis, so we concludeF(Dmn,δ,gG)′(∆)(Dmn,δ,gG(z))′≥ min|z|=1Fk(∆)k(z) = Fk(∆)k(1), (3.10)andη∗ = k(1) = 2η − 1 + (λ+ 2) (2− 2η)1∫0tλ+2t+ 1dt. Theorem 3.4. Let k belongs to C in ∆, k(0) = 1, and h(z) = k(z) + zk′(z). If f ∈ Aand satisfies the fuzzy differential subordinationF(Dmn,δ,gf)′(∆)(Dmn,δ,gf(z))′≤ Fh(∆)h(z)→(Dmn,δ,gf(z))′≺F h(z), (3.11)thenFDmn,δ,gf(∆)Dmn,δ,gf (z)z≤ Fk(∆)k(z)→Dmn,δ,gf (z)z≺F k(z). (3.12)The result is sharp.Proof. Forq(z) =Dmn,δ,gf (z)z=z +∞∑j=2[1 + (j − 1) cn(δ)]m ajbjzjz= 1 +∞∑j=2[1 + (j − 1) cn(δ)]m ajbjzj−1,we obtain thatq(z) + zq′(z) =(Dmn,δ,gf (z))′,158 Sheza M. El-Deeb and Alina Alb LupaşsoF(Dmn,δ,gf)′(∆)(Dmn,δ,gf(z))′≤ Fh(∆)h(z)impliesFq(∆)(q(z) + zq′(z))≤ Fh(∆)h(z) = Fk(∆)(k(z) + zk′(z)).Applying Lemma 2.3, we haveFq(∆)q(z) ≤ Fk(∆)k(z)→ FDmn,δ,gf(∆)Dmn,δ,gf (z)z≤ Fk(∆)k(z),and we getDmn,δ,gf (z)z≺F k(z).The result is sharp. Theorem 3.5. Consider h ∈ H(∆) with h(0) = 1, which satisfies<(1 +zh′′(z)h′(z))>−12.If f ∈ A and the fuzzy differential subordinationF(Dmn,δ,gf)′(∆)(Dmn,δ,gf(z))′≤ Fh(∆)h(z)→(Dmn,δ,gf(z))′≺F h(z), (3.13)holds, thenFDmn,δ,gf(∆)Dmn,δ,gf (z)z≤ Fk(∆)k(z) i.eDmn,δ,gf (z)z≺F k(z), (3.14)wherek(z) =1zz∫0h(t)dt ,the function k is convex and it is the fuzzy best dominant.Proof. Letq(z) =Dmn,δ,gf (z)z= 1 +∞∑j=2[1 + (j − 1) cn(δ)]m ajbjzj−1, q ∈ H [1, 1] ,where <(1 + zh′′(z)h′ (z))> −12 . From Lemma 2.1, we havek(z) =1zz∫0h(t)dtbelongs to the class C, which satisfies the fuzzy differential subordination (3.13). Sincek(z) + zk′(z) = h(z),it is the fuzzy best dominant.Fuzzy differential subordinations connected with convolution 159We haveq(z) + zq′(z) =(Dmn,δ,gf (z))′,then (3.13) becomesFq(∆)(q(z) + zq′(z))≤ Fh(∆)h(z).Applying Lemma 2.3, we haveFq(∆)q(z) ≤ Fk(∆)k(z), i.e. FDmn,δ,gf(∆)Dmn,δ,gf (z)z≤ Fk(∆)k(z),thenDmn,δ,gf (z)z≺F k(z). Putting h(z) = 1+(2β−1)z1+z in Theorem 3.5, we obtain the following corollary:Corollary 3.6. Let h = 1+(2β−1)z1+z a convex function in ∆, with h(0) = 1, 0 ≤ β < 1.If f ∈ A and verifies the fuzzy differential subordinationF(Dmn,δ,gf)′(∆)(Dmn,δ,gf(z))′≤ Fh(∆)h(z), i.e(Dmn,δ,gf(z))′≺F h(z),thenk(z) = 2β − 1 + 2 (1− β)zln (1 + z) ,the function k is convex and it is the fuzzy best dominant.Concluding, all the above results give us information about fuzzy differential sub-ordinations for the operator Dmn,δ,g, we give some properties for the class HFα,m (n, η)of univalent analytic functions.References[1] Alb Lupaş, A., On special fuzzy differential subordinations using convolution product ofSălăgean operator and Ruscheweyh derivative, J. Comput. Anal. Appl., 15(2013), no. 8,1-6.[2] Alb Lupaş, A., Oros, Gh., On special fuzzy differerential subordinations using Sălăgeanand Ruscheweyh operators, Appl. Math. Comput., 261(2015), 119-127.[3] Al-Oboudi, F.M., On univalent functions defined by a generalized Sălăgean operator, Int.J. Math. Math. Sci., 27(2004), 1429-1436.[4] Bulboacă, T., Differential Subordinations and Superordinations, Recent Results, Houseof Scientific Book Publ., Cluj-Napoca, 2005.[5] El-Deeb, S.M., Maclaurin coefficient estimates for new subclasses of bi-univalent func-tions connected with a q-analogue of Bessel function, Abstr. Appl. Anal., (2020), ArticleID 8368951, 1-7, https://doi.org/10.1155/2020/8368951.[6] El-Deeb, S.M., Alb Lupaş, A., Fuzzy differential subordinations associated with an inte-gral operator, An. Univ. Craiova Ser. Mat. Inform. , 27(2020), no. 1, 133–140.[7] El-Deeb, S.M., Bulboacă, T., Fekete-Szegö inequalities for certain class of analytic func-tions connected with q-anlogue of Bessel function, J. Egyptian Math. Soc., (2019), 1-11,https://doi.org/10.1186/s42787-019-0049-2.160 Sheza M. El-Deeb and Alina Alb Lupaş[8] El-Deeb, S.M., Bulboacă, T., Differential sandwich-type results for symmetric func-tions connected with a q-analog integral operator, Mathematics, 7(2019), no. 12, 1-17,https://doi.org/10.3390/math7121185.[9] El-Deeb, S.M., Oros, G., Fuzzy differential subordinations connected with the linear op-erator, Math. Bohem., 146(2021), no. 4, 397-406.[10] Gal, S.G., Ban, A.I., Elemente de Matematica Fuzzy, Editura Univ. din Oradea, 1996.[11] Miller, S.S., Mocanu, P.T., Differential Subordination: Theory and Applications, Serieson Monographs and Textbooks in Pure and Applied Mathematics, Vol. 225, MarcelDekker Inc., New York and Basel, 2000.[12] Oros, G.I., Oros, Gh., The notation of subordination in fuzzy sets theory, Gen. Math.,19(2011), no. 4, 97-103.[13] Oros, G.I., Oros, Gh., Fuzzy differential subordination, Acta Univ. Apulensis, 30(2012),55-64.[14] Oros, G.I., Oros, Gh., Dominant and best dominant for fuzzy differential subordinations,Stud. Univ. Babeş-Bolyai Math., 57(2012), no. 2, 239-248.[15] Sălăgean, G.S., Subclasses of univalent functions, Lecture Notes in Math., 1013, SpringerVerlag, Berlin, 1983, 362-372.[16] Srivastava, H.M., El-Deeb, S.M., A certain class of analytic functions of complex orderwith a q-analogue of integral operators, Miskolc Math. Notes, 21(2020), no. 1, 417–433.[17] Yousef, F., Al-Hawary, T., Murugusundaramoorthy, G., Fekete–Szegö functional prob-lems for some subclasses of bi-univalent functions defined by Frasin differential operator,Afr. Mat., 30(2019), 495-503, https://doi.org/10.1007/s13370-019-00662-7.Sheza M. El-DeebDepartment of Mathematics, Faculty of Science,Damietta University, New Damietta 34517, Egypt,andDepartment of Mathematics,College of Science and Arts in Al-Badaya,Qassim University, Buraidah, Saudi Arabiae-mail: shezaeldeeb@yahoo.com; s.eldeeb@qu.edu.saAlina Alb LupaşDepartment of Mathematics and Computer Science,University of Oradea,Str. Universităţii nr. 1, 410087 Oradea, Romaniae-mail: dalb@uoradea.ro

Fuzzy differential subordinations connected with convolution

Sheza M. El-Deeb

Alina Alb-Lupas

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Fuzzy differential subordinations connected with convolution

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