The purpose of the present paper is to establish some results involving coefficient conditions, distortion bounds, extreme points, convolution, and convex combinations for a new class of harmonic univalent functions in the open unit disc \ associated with the Salagean operator. We also discuss a class preserving integral operator. Relevant connections of the results presented here with various known results are briefly indicated

Darwish, H. E.

Lashin, A. Y.

Soileh, S. M.

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LXIX (2014) – Fasc. II, pp. 147–158doi: 10.4418/2014.69.2.13SUBCLASS OF HARMONIC STARLIKE FUNCTIONSASSOCIATED WITH SALAGEAN DERIVATIVEH. E. DARWISH - A. Y. LASHIN - S. M. SOILEHThe purpose of the present paper is to establish some results involv-ing coefficient conditions, distortion bounds, extreme points, convolution,and convex combinations for a new class of harmonic univalent functionsin the open unit disc associated with the Salagean operator. We alsodiscuss a class preserving integral operator. Relevant connections of theresults presented here with various known results are briefly indicated.1. IntroductionA continuous complex-valued function f = u+ iv is said to be harmonic in asimply connected domain D⊂ C if both u and v are real harmonic in D. Thereis a close inter-relation between analytic functions and harmonic functions. Forexample, for real harmonic functions u and v there exist analytic functions Uand V so that u =ℜ(U) and v = Im(V ). Thenf (z) = h(z)+g(z),where h and g are respectively, the analytic functions (U+V )/2 and (U−V )/2.In this case, the Jacobian of f = h+g is given byJ f =∣∣h′(z)∣∣2− ∣∣g′(z)∣∣2 .Entrato in redazione: 16 settembre 2013AMS 2010 Subject Classification: 30C45.Keywords: Harmonic, Univalent, Starlike, Convolution.148 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEHThe mapping z→ f (z) is orientation preserving and locally one-to-one inD if and only if J f > 0 in D. The function f = h+ g is said to be harmonicunivalent in D if the mapping z→ f (z) is orientation preserving, harmonic andone-to-one in D. We call h the analytic part and g the co-analytic part of f . SeeClunie and Sheil-Small [2].Denote by H the class of functions f = h+ g that are harmonic univalentand orientation preserving in the open unit disk U = {z : |z| < 1} for whichf (0) = fz(0)− 1 = 0. Then for f = h+ g ∈ H, we may express the analyticfunctions f and g ash(z) = z+∞∑k=2ak zk, g(z) =∞∑k=1bk zk, |b1|< 1. (1)Note that H reduces to the class S of normalized analytic univalent functionsif the co-analytic part of its member is zero. For this class the function f (z) maybe expressed asf (z) = z+∞∑k=2akzk. (2)A function f = h+ g with h and g be given by (1) is said to be harmonicstarlike of order β ,(0≤ β < 1) for |z|= r < 1, if∂∂θ(arg f (reiθ )) =ℜ{zh′(z)− zg′(z)h(z)+g(z)}≥ β .The class of all harmonic starlike functions of order β is denoted by S∗H(β )and extensively studied by Jahangiri [6]. The case β = 0 and b1 = 0 were studiedby Silverman and Silvia [19] and Silverman [18], for other related works of theclass H, (see also [1], [3], [4], [5], [7], [8], [9], [12], [15], [16]).Definition 1.1. Let Dn f = Dnh+Dng with h and g be given by (1).WhereDnh(z) = z+∞∑k=2knakzk, Dng(z) =∞∑k=1knbkzk, |b1|< 1. (3)Then f ∈ S∗H,n(α,β ) if and only if, for β ,(0≤ β < 1) and α ≥ 0,ℜ{αDn+2h(z)+(1−α)Dn+1h(z)+αDn+2g(z)+(α−1)Dn+1g(z)Dnh(z)+Dng(z)}≥ β .SUBCLASS OF HARMONIC STARLIKE FUNCTIONS 149We note that for α = 0 and n = 0, the class S∗H,n(α,β ) reduces to the classS∗H(β ). Also we note thatS∗H,0(α,β ) = S∗H(α,β )={f (z) ∈ H :ℜ{αz2h′′(z)+zh′(z)+αz2g′′(z)+(2α−1)zg′(z)h(z)+g(z)}≥ β}.Further, if the co-analytic part g(z) is zero the class S∗H,n(α,β ) reduces to theclass Pn(α,β ) of function f ∈ S which satisfy the conditionℜ{az2(Dn f )′′+ z(Dn f )′Dn f (z)}≥ βfor some β (0≤ β < 1),α ≥ 0, f (z)z 6= 0 and z ∈U.Observe that the classes P0(α,β ) and P0(α,0) were introduced and studiedby many authors such as Obradovic and Joshi [13], Padmanabhan [14], Liu andOwa [11], Xu and Yang [21], Singh and Gupta [20] and Lashin [10]. We alsonote that for α = 0, the class P0(0,β ) was studied by Silverman [17].2. Main ResultsWe begin with the statement of the following lemma due to Jahangiri [6].Lemma 2.1. Let f = h+g with h and g of the form (1). Furthermore, let∞∑k=2k−β1−β |ak |+∞∑k=1k+β1−β |bk | ≤ 1where 0 ≤ β < 1. Then f is harmonic, orientation preserving, univalent in U,and f ∈ S∗H(β ).Theorem 2.2. Let f = h+g with h and g of the form (1). If∞∑k=2knαk(k−1)+ k−β1−β |ak |+∞∑k=1knαk(k+1)+ k+β1−β |bk | ≤ 1 (4)for some β ,(0≤ β < 1) and α ≥ 0, then f is harmonic, orientation preserving,univalent in U and f ∈ S∗H,n(α,β ).Proof. Sincek−β1−β ≤ knαk(k−1)+ k−β1−β andk+β1−β ≤ knαk(k+1)+ k+β1−β ,(k ≥ 1),150 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEHit follows from Lemma 2.1 , that f ∈ S∗H(β ) and hence f is harmonic, ori-entation preserving and univalent in U . Now, we only need to show that if (4)holds thenℜ{αDn+2h(z)+(1−α)Dn+1h(z)+αDn+2g(z)+(α−1)Dn+1g(z)Dnh(z)+Dng(z)}=ℜA(z)B(z)≥ β .Using the fact thatℜ(w)≥ β if and only if |1−β +w| ≥ |1+β −w| , it sufficesto show that|A(z)+(1−β )B(z)|− |A(z)− (1+β )B(z)| ≥ 0, (5)whereA(z) = αDn+2h(z)+(1−α)Dn+1h(z)+αDn+2g(z)+(α−1)Dn+1g(z)andB(z) = Dnh(z)+Dng(z)Substituting for A(z) and B(z) in (5), we obtain|A(z)+(1−β )B(z)|− |A(z)− (1+β )B(z)|=∣∣∣∣∣(2−β )z+ ∞∑k=2 kn [(αk+1)(k−1)+2−β ]ak zk+∞∑k=1kn [αk(k+1)− (k−1+β )]bk zk∣∣∣∣∣−∣∣∣∣∣−β z+ ∞∑k=2 kn [(αk+1)(k−1)−β ]ak zk+∞∑k=1kn [αk(k+1)− (k+1+β )]bk zk∣∣∣∣∣≥ (2−β ) |z|−∞∑k=2kn [(αk+1)(k−1)+2−β ] |ak| |z|k−∞∑k=1kn |αk(k+1)− (k−1+β )| |bk| |z|k−β |z|−∞∑k=2kn [(αk+1)(k−1)−β ] |ak | |z|k−∞∑k=1kn |αk(k+1)− (k+1+β )| |bk | |z|kSUBCLASS OF HARMONIC STARLIKE FUNCTIONS 151≥ 2(1−β ) |z|−2∞∑k=2kn [αk(k−1)+ k−β ] |ak |∣∣zk∣∣−2∞∑k=1kn [αk(k+1)+(k+β )] |bk| |z|k≥ 2(1−β ) |z|{1−∞∑k=2kn [αk(k−1)+k−β ]1−β |ak |−∞∑k=1kn [αk(k+1)+k+β ]1−β |bk |}≥ 0,by the given condition (4).The Harmonic functionf (z) = z+∞∑k=21−βkn [αk(k−1)+ k−β ]xk zk +∞∑k=11−βkn [αk(k+1)+ k+β ]ykzk (6)where ∑∞k=2 |xk |+∑∞k=1 |yk |= 1, show that the coefficient bound given in Theo-rem 2.2 is sharp.The functions of the form (6) are in S∗H.n(α,β ) since∞∑k=2kn[αk(k−1)+ k−β ]1−β |ak |+∞∑k=1kn[αk(k+1)+ k+β ]1−β |bk |=∞∑k=2|xk |+∞∑k=1|yk |= 1.We note that:(i) when n = 0 and α = 0 in Theorem 2.2, we obtain the result obtained byJahangiri [[6], Theorem 1];(ii) when n = 0 and g(z) = 0 in Theorem 2.2, we obtain the result obtainedby Lashin [[10], Theorem 2.1].We denote by S∗H,n(α,β ) to the class of functions f ∈ S∗H,n(α,β ) whosecoefficients satisfy the condition (4).Theorem 2.3. Let 0≤ α1 < α2 and 0≤ β < 1. Then S∗H,n(α2,β )⊂ S∗H,n(α1,β ).Proof. It follows from (4) that∞∑k=2kn[α1k(k−1)+ k−β ]1−β |ak |+∞∑k=1kn[α1k(k+1)+ k+β ]1−β |bk |<∞∑k=2kn[α2k(k−1)+ k−β ]1−β |ak |+∞∑k=1kn[α2k(k+1)+ k+β ]1−β |bk | ≤ 1for f ∈ S∗H,n(α2,β ). Hence f ∈ S∗H,n(α1,β ).152 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEHTheorem 2.4. Let α > 0, 0≤ β < 1. ThenS∗H,n+1(α,β )⊂ S∗H,n(α,β )The proof follows immediately from Theorem 2.2.In the following Corollary we show that the functions in S∗H,n(α,β ) are star-like harmonic in U .Corollary 2.5. For 0≤ α and 0≤ β < 1,S∗H,n(α,β )⊂ S∗H(β ).The proof is now immediate from Theorems 2.3 and 2.4.3. Distortion Bounds and Extreme PointsNow we obtain the distortion bounds for functions in the class S∗H,n(α,β ).Theorem 3.1. Let f = h+ g with h and g of the form (1) and f ∈ S∗H,n(α,β ).Then for |z|= r < 1, we have| f (z)| ≤ (1+ |b1|)r+ 12n[1−β2α+2−β −2α+1+β2α+2−β |b1|]r2 (7)| f (z)| ≥ (1−|b1|)r− 12n[1−β2α+2−β −2α+1+β2α+2−β |b1|]r2 (8)where|b1| ≤ 1−β2α+1+β .The result is sharp.Proof. We shall prove the first inequality. Let f ∈ S∗H,n(α,β ). Then we have| f (z)| ≤ (1+ |b1|)r+∞∑k=2(|ak |+ |bk |)rk ≤ (1+ |b1|)r+ r2∞∑k=2(|ak |+ |bk |)= (1+ |b1|)r+ 1−β2n [2α+2−β ]∞∑k=22n [2α+2−β ]1−β (|ak |+ |bk |)r2and so| f (z)| ≤ (1+ |b1|)r+1−β2n [2α+2−β ]∞∑k=2kn[αk(k−1)+ k−β1−β |ak |+αk(k+1)+ k+β1−β |bk |]r2≤ (1+ |b1|)r+ 1−β2n [2α+2−β ][1− 2α+1+β1−β |b1|]r2= (1+ |b1|)r+ 12n[1−β2α+2−β −2α+1+β2α+2−β |b1|]r2.SUBCLASS OF HARMONIC STARLIKE FUNCTIONS 153The proof of the inequality (8) is similar, thus we omit it.The upper bound given for f ∈ S∗H,n(α,β )) is sharp and the equality occursfor the functionf (z) = z+ |b1|z+ 12n[1−β2α+2−β −2α+1+β2α+2−β |b1|]z2(z = r),|b1| ≤ 1−β2α+1+β .This completes the proof of Theorem 3.1.Now we determine the extreme points of the closed convex hull of the classS∗H,n(α,β ) denoted by clcoHS∗H,n(α,β ).Theorem 3.2. Let f = h + g, where h and g are given by (1). Then f ∈clcoHS∗H,n(α,β ) if and only iff (z) =∞∑k=1(Xk hk +Yk gk) (9)whereh1(z) = zhk(z) = z+1−βkn [αk(k−1)+ k−β ]zk (k = 2,3, . . .),gk(z) = z+1−βkn [αk(k+1)+ k+β ]zk (k = 1,2,3, . . .),∑∞k=1(Xk +Yk) = 1, Xk ≥ 0 and Yk ≥ 0. In particular, the extreme points of theclass S∗H,n(α,β ) are {hk} and {gk} respectively.Proof. For a function f of the form (9), we havef (z) =∞∑k=1(Xkhk +Ykgk)=∞∑k=1(Xk +Yk)z+∞∑k=21−βkn [αk(k−1)+ k−β ]Xkzk+∞∑k=11−βkn [αk(k+1)+ k+β ]Ykzk= z+∞∑k=21−βkn [αk(k−1)+ k−β ]Xkzk +∞∑k=11−βkn [αk(k+1)+ k+β ]Ykzk.154 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEHBut∞∑k=2kn [αk(k−1)+ k−β ]1−β[1−βkn [αk(k−1)+ k−β ]Xk]+∞∑k=2kn [αk(k+1)+ k+β ]1−β[1−βkn [αk(k+1)+ k+β ]Xk]=∞∑k=2Xk +∞∑k=1Yk = 1−X1 ≤ 1.Thus f ∈ clcoHS∗H,n(α,β ).Conversely, suppose that f ∈ clcoHS∗H,n(α,β ). SetXk =kn [αk(k−1)+ k−β ]1−β |ak | (k = 2,3, . . .), (10)andYk =kn [αk(k+1)+ k+β ]1−β |bk | (k = 1,2,3, . . .). (11)Then by the inequality (4), we have 0≤ Xk ≤ 1(k= 2,3, . . .) and 0≤Yk ≤ 1(k=1,2, . . .). Define X1 = 1−∑∞k=2 Xk −∑∞k=1Yk and note that X1 ≥ 0. Thus weobtain f (z) = ∑∞k=1 Xkhk +Ykgk . This completes the proof of Theorem 3.2.4. Convolution and Convex CombinationsFor two harmonic functionsf (z) = z+∞∑k=2ak zk +∞∑k=1bkzkandF(z) = z+∞∑k=2Ak zk +∞∑k=1Bk zkwe define their convolution( f ∗F)(z) = z+∞∑k=2ak Ak zk +∞∑k=1bk Bk zk,using this definition, we show that the class S∗H,n(α,β ) is closed under convolu-tion.Theorem 4.1. For some 0 ≤ β < 1 and α ≥ 0, let f ,F ∈ S∗H,n(α,β ). Thenf ∗F ∈ S∗H,n(α,β ).SUBCLASS OF HARMONIC STARLIKE FUNCTIONS 155Proof. We note that |Ak| ≤ 1 and |Bk| ≤ 1. Now for the convolution ( f ∗F) wehave∞∑k=2kn[αk(k−1)+ k−β ]1−β |Akak|+∞∑k=1kn[αk(k+1)+ k+β ]1−β |Bkbk|≤∞∑k=2kn[αk(k−1)+ k−β ]1−β |ak|+∞∑k=1kn[αk(k+1)+ k+β ]1−β |bk| ≤ 1.Therefore f ∗F ∈ S∗H,n(α,β ).We show that the class S∗H,n(α,β ) is closed under convex combination of itsmembers.Theorem 4.2. The class S∗H,n(α,β ) is closed under convex combination.Proof. For (i = 1,2,3 . . .) let fi ∈ S∗H,n(α,β ) where fi(z) is given byfi(z) = z+∞∑k=2akizk +∞∑k=1bkizk.Then by (4), we have∞∑k=2kn[αk(k−1)+ k−β ]1−β |aki|+∞∑k=1kn[αk(k+1)+ k+β ]1−β |bki| ≤ 1.For ∑∞i=1 ti = 1, 0≤ ti ≤ 1, the convex combination of fi may be written as∞∑i=1ti fi = z+∞∑k=2(∞∑i=1tiaki)zk +∞∑k=1(∞∑i=1tibki)zk.Then by (4), we have∞∑k=2kn[αk(k−1)+ k−β ]1−β∣∣∣∣∣ ∞∑i=1 tiaki∣∣∣∣∣+ ∞∑k=1 kn [αk(k+1)+ k+β ]1−β∣∣∣∣∣ ∞∑i=1 tibki∣∣∣∣∣≤∞∑i=1ti(∞∑k=2kn[αk(k−1)+ k−β ]1−β |aki|+∞∑k=1kn[αk(k+1)+ k+β ]1−β |bki|)≤∞∑i=1ti = 1.Therefore ∑∞i=1 ti fi ∈ S∗H,n(α,β ).156 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEH5. A Family of Class Preserving Integral OperatorLet f (z) = h(z)+g(z) ∈ H be given by(1) then F(z) defined by relationF(z) =c+1zc∫ z0tc−1h(t)dt+c+1zc∫ z0tc−1g(t)dt , (c >−1) (12)Theorem 5.1. Let f = h+ g, where h and g are given by (1) be in the classS∗H,n(α,β ), then F(z) defined by (12) also belong to the class S∗H,n(α,β ).Proof. From the representation of F , it follows thatF(z) = z+∞∑k=2c+1c+ kak zk +∞∑k=1c+1c+ kbkzk.Now∞∑k=2kn[αk(k−1)+ k−β ]1−β(c+1c+ k|ak |)+∞∑k=1kn[αk(k+1)+ k+β ]1−β(c+1c+ k|bk|)≤∞∑k=2kn[αk(k−1)+ k−β ]1−β |ak|+∞∑k=1kn[αk(k+1)+ k+β ]1−β |bk| ≤ 1by (4). Thus F(z) ∈ S∗H,n(α,β ).AcknowledgementsThe authors would like to thank the referee for his/her careful reading and mak-ing some valuable comments which have essentially improved the presentationof this paper.REFERENCES[1] O. P. Ahuja - J. M. Jahangiri, A subclass of harmonic univalent functions, J. ofNat. Geom. 20 (2001), 45–56.[2] J. Clunie - T. Sheil-Small, Harmonic univalent functions, Ann. Acad. Sci. Fen.Series AI Math. 9 (3) (1984), 3–25.[3] R. M. El-Ashwah - M. K. Aouf - A. M. Hassan - A. H. Hassan, A Unified represen-tation of some starlike and convex harmonic functions with negative coefficients,Opuscula Math. 33 (2) (2013), 273–281.SUBCLASS OF HARMONIC STARLIKE FUNCTIONS 157[4] B. A. Frasin, Comprehensive family of harmonic univalent functions, SUTJ. Math.42 (1) (2006), 145–155.[5] J. M. Jahangiri, Coefficient bounds and univalence criteria for harmonic functionswith negative coefficients, Ann. Univ. Mariae Curie-Sklodowska Sect. A 52 (2)(1998), 57–66.[6] J. M. Jahangiri, Harmonic functions starlike in the unit disc, J. Math. Anal. Appl.235 (1999), 470–477.[7] J. M. Jahangiri - G. Murugusundaramoorthy - K. Vijaya, Starlikeness of harmonicfunctions defined by Ruscheweyh derivatives, Indian Acad. Math. 26 (1) (2004),191–200.[8] A. Janteng - S. Abdul Halim, Properties of harmonic functions which are convexof order β with respect to symmetric points, Tamkang J. Math. 40 (1) (2009),31–39.[9] Y. C. Kim - J. M. Jahangiri - J. H. Choi, Certain convex harmonic functions, Int. J.Math. Math. Sci. 29 (2002), 459–465.[10] A. Y. Lashin, On a certain subclass of starlike functions with negative coefficients,J. Ineq. Pure. Appl. Math. 10 (2) (2009), 1–18.[11] J. L. Liu - S. Owa, Sufficient conditions for starlikeness, Indian J. Pure Appl. Math.Soc. 33 (2002), 313–318.[12] N. Magesh - S. Mayilvaganan, On a Subclass of Harmonic Convex Functionsof Complex Order, Int. J. Math. Math. Sci. 2012 (2012), Article ID 496731, 13pages.[13] M. Obradovic - S. B. Joshi, On certain classes of strongly starlike functions, Tai-wanese J. Math. 2 (3) (1998), 297–302.[14] K. S. Padmanabhan, On sufficient conditions for starlikeness, Indian J. Pure Appl.Math. 32 (4) (2001), 543–550.[15] S. Ponnusamy - A. Rasila, Plannar harmonic mappings, RMS MathematicsNewsletter. 17 (2) (2007), 40–57.[16] S. Ponnusamy - A. Rasila, Planar harmonic and quasiconformal mappings, RMSMathematics Newsletter 17 (3) (2007), 85–101.[17] H. Silverman, Partial sums of starlike and convex functions, J. Math. Anal. Appl.209 (1997), 221–227.[18] H. Silverman, Harmonic univalent functions with negative coefficients, J.Math.Anal. Appl. 220 (1998), 283–289.[19] H. Silverman - E. M. Silvia, Subclasses of harmonic univalent functions, NewZealand J. Math. 28 (1999), 275–284.[20] S. Singh - S. Gupta, First order differential subordinations and starlikeness ofanalytic maps in the unit disc, Kyungpook Math. J. 45 (2005), 395–404.[21] N. Xu - D. Yang, Some criteria for starlikeness, and strongly starlikeness, Bull.Korean Math. Soc. 42 (3) (2005), 579–590.158 H. E. DARWISH - A. Y. LASHIN - S. M. SOILEHH. E. DARWISHDepartment of Mathematics Faculty of ScienceMansoura University, Mansoura 35516, Egypte-mail: Darwish333@yahoo.comA. Y. LASHINDepartment of Mathematics Faculty of ScienceMansoura University, Mansoura 35516, Egypte-mail: aylashin@mans.edu.egS. M. SOILEHDepartment of Mathematics Faculty of ScienceMansoura University, Mansoura 35516, Egypte-mail: s soileh@yahoo.com

Subclass of harmonic starlike functions associated with salagean derivative

H. E. Darwish

A. Y. Lashin

S. M. Soileh

Directory of Open Access Journals

http://doaj.org/search?source=%7B%22query%22%3A%7B%22bool%22%3A%7B%22must%22%3A%5B%7B%22term%22%3A%7B%22id%22%3A%225622d7e7965c47c5bb72778b505cf0e2%22%7D%7D%5D%7D%7D%7D

Subclass of harmonic starlike functions associated with salagean derivative

Abstract

Similar works

Full text

Available Versions

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

Directory of Open Access Journals