In this paper, we define and investigate a subclass of univalent harmonic functions defined by Salagean integral operator with respect to symmetric points. We obtain coefficient conditions, extreme points, distortion bounds, convex combinations for this family of harmonic univalent functions

Aouf, M. K.

El-Ashwah, R. M.

El-Deeb, Sheza

Shamandy, A.

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

LE MATEMATICHEVol. LXVII (2012) – Fasc. II, pp. 203–216doi: 10.4418/2012.67.2.15UNIVALENT HARMONIC FUNCTIONS DEFINED BYSALAGEAN INTEGRAL OPERATOR WITH RESPECT TOSYMMETRIC POINTSMOHAMED K. AOUF - RABHA M. EL-ASHWAHALI SHAMANDY - SHEZA M. EL-DEEBIn this paper, we define and investigate a subclass of univalent har-monic functions defined by Salagean integral operator with respect tosymmetric points. We obtain coefficient conditions, extreme points, dis-tortion bounds, convex combinations for this family of harmonic univalentfunctions.1. IntroductionA continuous complex-valued function f = u+ iv defined in a simply-connectedcomplex domain D is said to be harmonic in D if both u and v are real harmonicin D. In any simply-connected domain we can writef = h+g , (1)where h and g are analytic in D. We call h the analytic part and g the co-analyticpart of f . A necessary and sufficient condition for f to be locally univalent andsense-preserving in D is that |h′(z)|> |g′(z)| in D (see [5]).Denote by SH the class of functions f of the form (1) that are harmonicunivalent and sense-preserving in the unit disc U = {z : |z| < 1} for whichEntrato in redazione: 23 marzo 2012AMS 2010 Subject Classification: 30C45.Keywords: Harmonic univalent functions, Integral operator, Symmetric points.204 M. K. AOUF - R. M. EL-ASHWAH - A. SHAMANDY - S. M. EL-DEEBf (0) = fz(0)− 1 = 0. Then for f = h+ g ∈ SH we may express the analyticfunctions h and g ash(z) = z+∞∑k=2ak zk , g(z) =∞∑k=1bkzk, |b1|< 1. (2)In 1984 Clunie and Shell-Small [6] investigated the class SH as well as itsgeometric subclasses and obtained some coefficient bounds. Since then, therehave been several related papers on SH and its subclasses.For f = h+g given by (2), we define the modified integral Salagean operatorof f asIm f (z) = Imh(z)+(−1)m Img(z) (m ∈ N0 = N∪{0}, N= {1,2, ...}) , (3)whereImh(z) = z+∞∑k=2k−m ak zk and Img(z) =∞∑k=1k−m bk zk.The integral operator Im was introduced by Salagean [10](see also [2], [3] and[7]).Definition 1.1. For 0≤ α < 1, m ∈ N0 and z = reiθ ∈U , let SHs(m;α) denotethe family of harmonic functions f of the form 2 such thatRe{2Im f (z)Im+1 f (z)− Im+1 f (−z)}> α, (4)where Im f is defined by (3).We denote by SHs(m;α) the subclasses of harmonic functions fm = h+gmin SHs(m;α) such that h and gm are of the formh(z) = z−∞∑k=2akzk, gm(z) = (−1)m∞∑k=1bkzk, |b1|< 1. (5)Remark 1.2. If the co-analytic part of f = h+g is zero, then SHs(−1;α) turnsout to be the class S∗s (α) of starlike functions with respect to symmetric pointswhich was introduced by Sakaguchi [9]. Also, SHs(−2,α) turns out to be theclass Ks(α) of convex functions with respect to symmetric points which wasintroduced by Das and Singh [8].Also, we note that SHs(−n−1,α) = SHs(n,α) and SHs(−n−1,α) = SHs(n,α)(n ∈ N0), which were studied by AL-Khal and Al-Kharsani [1].In this paper, we extend the above results to the classes SHs(m;α) andSHs(m;α). We also obtain coefficient conditions, extreme points, distortionbounds, convolution conditions and convex combinations for SHs(m;α).UNIVALENT HARMONIC FUNCTIONS . . . 2052. Coefficient characterizationUnless otherwise mentioned, we assume throughout this paper that m ∈ N0,a1 = 1 and 0≤ α < 1. We begin with a sufficient condition for functions in theclass SHs(m;α).Theorem 2.1. Let f = h+g be such that h and g are given by (2). Furthermore,let∞∑k=1{(2k−1)−m−1 (2k−1−α) |a2k−1|+(2k)−m |a2k|+(2k−1)−m−1 (2k−1+α) |b2k−1|+(2k)−m |b2k|}≤ 2(1−α) . (6)Then f is sense-preserving, harmonic univalent in U and f ∈ SHs(m;α).Proof. Note that f is sense-preserving in U . Now we show that f ∈ SHs(m;α).We only need to show that if (6) holds then the condition (4) is satisfied.Using the fact that Rew > α if and only if |1−α +w| > |1+α −w|, itsuffices to show that∣∣2Im f (z)+(1−α)[ Im+1 f (z)− Im+1 f (−z)]∣∣− ∣∣2Im f (z)− (1+α)[Im+1 f (z)− Im+1 f (−z)]∣∣> 0 (7)Substituting for Im f (z) and Im+1 f (z) in (7) yields, by using (6) and 0≤ α < 1,we obtain∣∣∣∣∣2(2−α)z+ ∞∑k=2 k−m−1[2k+(1−α)(1− (−1)k)]ak zk−(−1)m+1∞∑k=1k−m−1[2k− (1−α)(1− (−1)k)]bk zk∣∣∣∣∣−∣∣∣∣∣−2α z+ ∞∑k=2 k−m−1[2k− (1+α)(1− (−1)k)]ak zk−(−1)m+1∞∑k=1k−m−1[2k+(1+α)(1− (−1)k)]bk zk∣∣∣∣∣=∣∣∣∣∣2(2−α)z+2 ∞∑k=2{(2k−2)−m a2k−2 z2k−2+(2k−1)−m−1 [2k−1−α+1]a2k−1 z2k−1}−206 M. K. AOUF - R. M. EL-ASHWAH - A. SHAMANDY - S. M. EL-DEEB2(−1)m+1∞∑k=1{(2k)−m b2k z2k +(2k−1)−m−1 [2k−1+α−1]b2k−1 z2k−1}∣∣∣∣∣−∣∣∣∣∣−2αz+2 ∞∑k=2{(2k−2)−m a2k−2 z2k−2+(2k−1)−m−1 [2k−1−α−1]a2k−1 z2k−1}−2(−1)m+1∞∑k=1{(2k)−m b2k z2k +(2k−1)−m−1 [2k−1+α+1]b2k−1 z2k−1}∣∣∣∣∣≥ 4(1−α)|z|−4∞∑k=2{(2k−2)−m |a2k−2| |z|2k−2+(2k−1)−m−1 [2k−1−α] |a2k−1| |z|2k−1}−4∞∑k=1{(2k)−m |b2k| |z|2k +(2k−1)−m−1 [2k−1+α] |b2k−1| |z|2k−1}≥ 4(1−α)−4[∞∑k=2{(2k−2)−m |a2k−2| +(2k−1)−m−1 [2k−1−α] |a2k−1|}+∞∑k=1{(2k)−m |b2k|+(2k−1)−m−1 [2k−1+α] |b2k−1|}].This last expression is non-negative by (6).The harmonic univalent functionsf (z) = z+∞∑k=2{1−α(2k−2)−m X2k−2 z2k−2+1−α(2k−1)−m−1 (2k−1−α)X2k−1 z2k−1}+∞∑k=1{1−α(2k)−mY2k z2k +1−α(2k−1)−m−1 (2k−1+α)Y2k−1 z2k−1}, (8)where∞∑k=2(|X2k−2|+ |X2k−1|) +∞∑k=1(|Y2k|+ |Y2k−1|) = 1, show that the coeffi-cient bound given by (6) is sharp. The functions of the form (8) are in SHs(m;α)because∞∑k=2{(2k−2)−m1−α |a2k−2|+(2k−1)−m−1 (2k−1−α)1−α |a2k−1|}+∞∑k=1{(2k)−m1−α |b2k|+(2k−1)−m−1 (2k−1+α)1−α |b2k−1|}UNIVALENT HARMONIC FUNCTIONS . . . 207=∞∑k=2(|X2k−2|+ |X2k−1|)+∞∑k=1(|Y2k|+ |Y2k−1|) = 1.This completes the proof.Putting m=−n−1 (n ∈ N0) in Theorem 2.1, we obtain the following corol-lary, correcting the results obtained by AL-Khal and Al-Kharsani [1, Theorem2.1].Corollary 2.2. Let f = h+g with h and g given by (2). Furthermore, let∞∑k=1{(2k−1)n (2k−1−α) |a2k−1|+(2k)n+1 |a2k|+(2k−1)n (2k−1+α) |b2k−1|+(2k)n+1 |b2k|}≤ 2(1−α) .Then f is sense-preserving, harmonic univalent in U and f ∈ SHs(n,α).In the following theorem, it is shown that the condition (6) is also necessaryfor functions fm = h+gm, where h and gm are of the form (5).Theorem 2.3. Let fm = h+ gm be such that h and gm are given by (5). Thenfm ∈ SHs(m;α) if and only if∞∑k=1{(2k)−m |a2k|+(2k−1)−m−1 (2k−1−α) |a2k−1|+(2k)−m |b2k|+(2k−1)−m−1 (2k−1+α) |b2k−1|}≤ 2(1−α) . (9)Proof. Since SHs(m;α)⊂ SHs(m;α), we only need to prove the “only if” partof the theorem. To this end, for functions fm of the form (5), we notice that thecondition Re{2Im fm(z)Im+1 fm(z)− Im+1 fm(−z)}> α is equivalent toRe2(1−α)z−∞∑k=2k−m−1[2k−α(1−(−1)k)]akzk−(−1)2(m+1)∞∑k=1k−m−1[2k+α(1−(−1)k)]bk zk2z−∞∑k=2k−m−1(1−(−1)k)akzk+(−1)2(m+1)∞∑k=1k−m−1(1−(−1)k)bk zk> 0which implies thatRe(1−α)z−∞∑k=2{(2k−2)−m a2k−2z2k−2+(2k−1)−m−1 [2k−1−α]a2k−1z2k−1}z−∞∑k=2(2k−1)−m−1 a2k−1z2k−1+(−1)2(m+1)∞∑k=1(2k−1)−m−1 b2k−1 z2k−1208 M. K. AOUF - R. M. EL-ASHWAH - A. SHAMANDY - S. M. EL-DEEB−(−1)2(m+1)∞∑k=1{(2k)−m b2k z2k+(2k−1)−m−1 [2k−1+α]b2k−1 z2k−1}z−∞∑k=2(2k−1)−m−1 a2k−1z2k−1+(−1)2(m+1)∞∑k=1(2k−1)−m−1 b2k−1 z2k−1> 0(10)The above required condition (10) must hold for all values of z in U . Uponchoosing the values of z on the positive real axis where 0≤ z = r < 1, we musthave(1−α)−∞∑k=2{(2k−2)−m a2k−2r2k−3+(2k−1)−m−1 [2k−1−α]a2k−1r2k−2}1−∞∑k=2(2k−1)−m−1 a2k−1r2k−2+∞∑k=1(2k−1)−m−1 b2k−1 r2k−2(11)−∞∑k=1{(2k)−m b2k r2k−1+(2k−1)−m−1 [2k−1+α]b2k−1 r2k−2}1−∞∑k=2(2k−1)−m−1 a2k−1r2k−2+∞∑k=1(2k−1)−m−1 b2k−1 r2k−2> 0.If the condition (9) does not hold, then the numerator in (11) is negative for rsufficiently close to 1. Hence there exists z0 = r0 in (0,1) for which the quotientin (11) is negative. This contradicts the required condition for fm ∈ SHs(m;α)and so the proof of Theorem 2 is completed.Putting m =−n−1 (n ∈ N0) in Theorem 2, we obtain the following corol-lary, correcting the result obtained by AL-Khal and Al-Kharsani [1, Theorem2.2].Corollary 2.4. Let fn = h+gn be given by (5). Then fn ∈ SHs(n,α) if and onlyif∞∑k=1{(2k−1)n (2k−1−α) |a2k−1|+(2k)n+1 |a2k|+(2k−1)n (2k−1+α) |b2k−1|+(2k)n+1 |b2k|}≤ 2(1−α) .3. Extreme points and distortion theoremOur next theorem is on the extreme points of convex hulls of SHs(m;α) denotedby clco SHs(m;α).UNIVALENT HARMONIC FUNCTIONS . . . 209Theorem 3.1. Let fm = h+ gm be such that h and gm are given by (5). Thenfm ∈ SHs(m;α) if and only iffm(z) =∞∑k=1[(X2k−1 h2k−1(z)+X2k h2k(z))+(Y2k−1gm2k−1(z)+Y2kgm2k(z))],(12)where h1(z) = z,h2k−1(z) = z− 1−α(2k−1)−m−1 (2k−1−α) z2k−1 (k ≥ 2),h2k−2(z) = z− 1−α(2k−2)−m z2k−2 (k ≥ 2),andgm2k−1(z) = z+(−1)m1−α(2k−1)−m−1 (2k−1+α) z2k−1 (k ≥ 1) ,gm2k(z) = z+(−1)m1−α(2k)−mz2k (k ≥ 1) ,Xk ≥ 0, Yk ≥ 0 ,∞∑k=1(X2k−1+X2k +Y2k−1+Y2k) = 1 .In particular, the extreme points of SHs(m;α) are {h2k−1} , {h2k−2} , {gm2k−1}and {gm2k}.Proof. For functions fm of the form (12), we havefm(z)= z−∞∑k=2{1−α(2k−2)−m X2k−2 z2k−2+1−α(2k−1)−m−1 (2k−1−α)X2k−1 z2k−1}+(−1)m∞∑k=1{1−α(2k)−mY2kz2k +1−α(2k−1)−m−1 (2k−1+α) Y2k−1 z2k−1}.Then∞∑k=2{(2k−2)−m1−α .(1−α(2k−2)−m X2k−2)+ (2k−1)−m−1(2k−1−α)1−α .(1−α(2k−1)−m−1(2k−1−α)X2k−1)}210 M. K. AOUF - R. M. EL-ASHWAH - A. SHAMANDY - S. M. EL-DEEB+∞∑k=1{(2k)−m1−α .(1−α(2k)−m Y2k)+ (2k−1)−m−1(2k−1+α)1−α .(1−α(2k−1)−m−1(2k−1+α) Y2k−1)}=∞∑k=2(X2k−2+X2k−1)+∞∑k=1(Y2k +Y2k−1) = 1−X1 ≤ 1and so fm ∈ SHs(m;α).Conversely, if fm ∈ clco SHs(m;α). SettingX2k−1 =(2k−1)−m−1 (2k−1−α)1−α a2k−1, X2k−2 =(2k−2)−m1−α a2k (k ≥ 2),andY2k−1 =(2k−1)−m−1 (2k−1+α)1−α b2k−1, Y2k−2 =(2k)−m1−α b2k (k ≥ 1).We obtainfm(z) =∞∑k=1[(X2k−1 h2k−1(z)+X2k h2k(z))+(Y2k−1gm2k−1(z)+Y2kgm2k(z))]as required.The following theorem gives the distortion bounds for functions in the classSHs(m;α) which yields a covering result for this class.Theorem 3.2. Let fm = h+gm be such that h and gm are given by (5) be in theclass SHs(m;α). Then for |z|= r < 1 we have| fm(z)| ≤ (1+ |b1|)r+ 12−m−1{1−α2− 1+α2|b1|}r2,and| fm(z)| ≥ (1−|b1|)r− 12−m−1{1−α2− 1+α2|b1|}r2Proof. We only prove the right-hand inequality. The proof for the left-handinequality is similar and will be omitted. Let fm(z) ∈ SHs(m;α). Taking theUNIVALENT HARMONIC FUNCTIONS . . . 211absolute value of fm we have| fm(z)| ≤ (1+ |b1|)r+∞∑k=2(|ak|+ |bk|)rk≤ (1+ |b1|)r+ r2∞∑k=2(|ak|+ |bk|)= (1+ |b1|)r+ (1−α)2−m∞∑k=22−m1−α (|ak|+ |bk|)r2≤ (1+ |b1|)r+ 12−m−1[1−α2− 1+α2|b1|]r2.The bounds given in Theorem 3.2 for functions fm = h+ gm of form (5) alsohold for functions of the form (2) if the coefficient condition (6) is satisfied.The upper bound given for fm ∈ SHs(m;α) is sharp and the equality occurs forthe functionsfm(z) = z+b1z +12−m−1[1−α2− 1+α2b1]z2,andfm(z) = z−b1z − 12−m−1[1−α2− 1+α2b1]z2show that the bounds given in Theorem 3.2 are sharp.The following covering result follows from the left hand inequality in The-orem 3.2.Corollary 3.3. Let the functions fm defined by (5) belong to the class SHs(m;α).Then {w : |w|< 2−m−1+α2−m− 2−m−1+α2−m|b1|}⊂ fm(U).4. Convolution and convex combinationFor our next theorem, we need to define the convolution of two harmonic func-tions. For harmonic functions of the form:fm(z) = z−∞∑k=2akzk +(−1)m∞∑k=1bkzk, |b1|< 1 (13)andFm(z) = z−∞∑k=2Ak zk +(−1)m∞∑k=1Bkzk (Ak ≥ 0; Bk ≥ 0) (14)212 M. K. AOUF - R. M. EL-ASHWAH - A. SHAMANDY - S. M. EL-DEEBwe define the convolution of two harmonic functions fm and Fm as( fm ∗Fm)(z) = fm(z)∗Fm(z) = z−∞∑k=2akAkzk +(−1)m∞∑k=1bk Bkzk. (15)Using this definition, we show that the class SHs(m;α) is closed under convo-lution.Theorem 4.1. For 0 ≤ β ≤ α < 1, let fm ∈ SHs(m;α) and Fm ∈ SHs(m;β ).Then fm ∗Fm ∈ SHs(m;α)⊂ SHs(m;β ).Proof. Let the function fm(z) defined by (13) be in the class SHs(m;α) and letthe function Fm(z) defined by (14) be in the class SHs(m;β ). Then the convolu-tion fm ∗Fm is given by (15). We wish to show that the coefficients of fm ∗Fmsatisfy the required condition given in Theorem 2.3. For Fm ∈ SHs(m;β ) wenote that 0≤ Ak ≤ 1 and 0≤ Bk ≤ 1. Now, for the convolution function fm ∗Fmwe obtain∞∑k=2{(2k−2)−m1−β |a2k−2|A2k−2+(2k−1)−m−1 (2k−1−β )1−β |a2k−1|A2k−1}+∞∑k=1{(2k)−m1−β |b2k|B2k +(2k−1)−m−1 (2k−1+β )1−β |b2k−1|B2k−1}≤∞∑k=2{(2k−2)−m1−β |a2k−2|+(2k−1)−m−1 (2k−1−β )1−β |a2k−1|}+∞∑k=1{(2k)−m1−β |b2k|+(2k−1)−m−1 (2k−1+β )1−β |b2k−1|}≤∞∑k=2{(2k−2)−m1−α |a2k−2|+(2k−1)−m−1 (2k−1−α)1−α |a2k−1|}+∞∑k=1{(2k)−m1−α |b2k|+(2k−1)−m−1 (2k−1+α)1−α |b2k−1|}≤ 1.since 0 ≤ β ≤ α < 1 and fm ∈ SHs(m;α). Therefore fm ∗Fm ∈ SHs(m;α) ⊂SHs(m;β ).Now, we show that the class SHs(m;α) is closed under convex combinationsof its members.Theorem 4.2. The class SHs(m;α) is closed under convex combination.UNIVALENT HARMONIC FUNCTIONS . . . 213Proof. For i = 1,2, ..., let fmi ∈ SHs(m;α), where fmi is given byfmi(z) = z−∞∑k=2aki zk +(−1)m∞∑k=1bki zk (|b1i |< 1; z ∈U).Then by using Theorem 2.3, we have∞∑k=2{(2k−2)−m1−α |ai,2k−2|+(2k−1)−m−1 (2k−1−α)1−α |ai,2k−1|}+∞∑k=1{(2k)−m1−α |bi,2k|+(2k−1)−m−1 (2k−1+α)1−α |bi,2k−1|}≤ 1. (16)For∞∑i=1ti = 1,0≤ ti ≤ 1, the convex combination of fmi may be written as∞∑i=1ti fmi(z) = z−∞∑k=2(∞∑i=1ti |aki |)zk +(−1)m∞∑k=1(∞∑i=1ti |bki |)zk. (17)Then, by using (16), we have∞∑k=2{(2k−2)−m1−α(∞∑i=1ti |ai,2k−2|)+(2k−1)−m−1 (2k−1−α)1−α(∞∑i=1ti |ai,2k−1|)}+∞∑k=1{(2k)−m1−α(∞∑i=1ti |bi,2k|)+(2k−1)−m−1 (2k−1+α)1−α(∞∑i=1ti |bi,2k−1|)}=∞∑i=1ti[∞∑k=2{(2k−2)−m1−α |ai,2k−2|+(2k−1)−m−1 (2k−1−α)1−α |ai,2k−1|}+∞∑k=1{(2k)−m1−α |bi,2k|+(2k−1)−m−1 (2k−1+α)1−α |bi,2k−1|}]≤∞∑i=1ti = 1,this is the condition required by (9) and so∞∑i=1ti fmi(z) ∈ SHs(m;α). This com-pletes the proof of Theorem 6.214 M. K. AOUF - R. M. EL-ASHWAH - A. SHAMANDY - S. M. EL-DEEB5. Properties of an integral operatorFinally, we study properties of an integral operator.Theorem 5.1. Let the functions fm(z) defined by (5) be in the class SHs(m;α)and let c be a real number such that c >−1. Then the function Fm(z) defined byFm(z) =c+1zcz∫0tc−1 fm(t)dt (18)belongs to the class SHs(m;α).Proof. From the representation of Fm(z), it follows thatFm(z) =c+1zcz∫0tc−1{h(t)+gm(t)}dt=c+1zc z∫0tc−1(t−∞∑k=2ak tk)dt+(−1)mz∫0tc−1(∞∑k=1bk tk)dt=c+1zc z∫0tcdt−∞∑k=2akz∫0tc+k−1dt+(−1)m∞∑k=1bkz∫0tc+k−1dt= z−∞∑k=2Akzk +(−1)m∞∑k=1Bkzk,where Ak =c+1c+ kak, Bk =c+1c+ kbk. Therefore∞∑k=2{(2k−2)−m A2k−2+(2k−1)−m−1 (2k−1−α)A2k−1}+∞∑k=1{(2k)−m B2k +(2k−1)−m−1 (2k−1+α)B2k−1}UNIVALENT HARMONIC FUNCTIONS . . . 215=∞∑k=2{(2k−2)−m ( c+1c+2k−2)a2k−2+(2k−1)−m−1 (2k−1−α)( c+1c+2k−1)a2k−1}+∞∑k=1{(2k)−m( c+1c+2k)b2k +(2k−1)−m−1 (2k−1+α)( c+1c+2k−1)b2k−1}≤∞∑k=2{(2k−2)−m a2k−2+(2k−1)−m−1 (2k−1−α)a2k−1}+∞∑k=1{(2k)−m b2k +(2k−1)−m−1 (2k−1+α)b2k−1}≤ 1−α.Since fm(z) ∈ SHs(m;α), therefore by Theorem 2.3 Fm(z) ∈ SHs(m;α).Remark 5.2. Putting m = −n− 1 (n ∈ N0) in Theorems 3.1, 3.2, 4.1, 4.2 and5.1, we obtain the results obtained by AL-Khal and Al-Kharsani [1] in Theorems2.7, 2.3, 2.5, 2.6, and 2.8, respectively.REFERENCES[1] R. A. Al-Khal - H. A. Al-Kharsani, Salagean-type harmonic univalent functionswith respect to symmetric points, The Australian J. Math. Analysis Appl. 4 (1)(2007), 1–13.[2] M. K. Aouf - F. M. Al-Oboudi - M. M. Hadain, An application of certain integraloperator, Math. (Cluj), 47 (70) (2) (2005), 121–124.[3] M. K. Aouf, Some applications of Salagean integral operator, Studia Univ. Babes-Bolyai Math. 55 (1) (2010), 21–30.[4] M. K. Aouf, The Salagean integral operator and strongly starlike functions, Stu-dia Univ. Babes-Bolyai Math., 56 (1) (2011), 109–115.[5] Y. Avci - E. Zlotkiewicz, On harmonic univalent mappings, Ann. Univ. Mariae-Curie Sklodowska Sect. A, 44 (1990), 1–7.[6] J. Clunie - T. Shell-Small, Harmonic univalent functions, Ann. Acad. Sci. Fenn.Ser. A. I. Math. 9 (1984), 3–25.[7] L. I. Cotirla, Harmonic univalent functions defined by an integral operator, ActaUniversitatis Apulensis, 17 (2009), 95–105.[8] R. N. Das - P. Singh, On subclasses of Schlicht mapping, Indian J. Pure Appl.Math. 8 (1977), 864–872.[9] K. Sakaguchi, On a certain univalent mapping, J. Math. Soc. Japan, 11 (1959),72–75.216 M. K. AOUF - R. M. EL-ASHWAH - A. SHAMANDY - S. M. EL-DEEB[10] G. S. Salagean, Subclasses of univalent functions, Lecture Notes in Math. 1013,Springer-Verlag 1983, 362-372.MOHAMED K. AOUFDepartment of Mathematics, Faculty of ScienceMansoura UniversityMansoura 35516, Egypte-mail: mkaouf127@yahoo.comRABHA M. EL-ASHWAHDepartment of Mathematics, Faculty of Science at DamiettaMansoura UniversityNew Damietta 34517, Egypte-mail: r elashwah@yahoo.comALI SHAMANDYDepartment of Mathematics, Faculty of ScienceMansoura UniversityMansoura 35516, Egypte-mail: shamandy16@hotmail.comSHEZA M. EL-DEEBDepartment of Mathematics, Faculty of Science at DamiettaMansoura UniversityNew Damietta 34517, Egypte-mail: shezaeldeeb@yahoo.com

Univalent harmonic functions defined by Salagean integral operator with respect to symmetric points

Sheza El-Deeb

M. K. Aouf

R. M. El-Ashwah

A. Shamandy

Directory of Open Access Journals

https://lematematiche.dmi.unict.it/index.php/lematematiche/article/download/923/835/

Univalent harmonic functions defined by Salagean integral operator with respect to symmetric points

Abstract

Similar works

Full text

Available Versions

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

Directory of Open Access Journals