In this paper, by making use of convolution, we obtain some interesting results for certain family of meromorphic p-valent functions defined by new linear operator.   

Aouf, M. K.

Mostafa, Adela

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

LE MATEMATICHEVol. LXIX (2014) – Fasc. II, pp. 259–266doi: 10.4418/2014.69.2.22ON CONVOLUTION PROPERTIES FOR CERTAINCLASSES OF p-VALENT MEROMORPHIC FUNCTIONSDEFINED BY LINEAR OPERATORADELA O. MOSTAFA - MOHAMED K. AOUFIn this paper, by making use of convolution, we obtain some interest-ing results for certain family of meromorphic p-valent functions definedby new linear operator.1. IntroductionLet Σp denote the class of functions of the form:f (z) = z−p+∞∑k=1ak−pzk−p (p ∈ N= {1,2, . . .}) , (1)which are analytic and p-valent in the punctured unit disc U∗ = {z : z ∈ C and0 < |z| < 1}. If f (z) and g(z) are analytic in U = U∗ ∪{0}, we say that f (z)is subordinate to g(z), written f ≺ g or f (z) ≺ g(z) (z ∈U), if there exists aSchwarz function w(z) in U with w(0) = 0 and |w(z)| < 1, such that f (z) =g(w(z)), (z ∈ U). Furthermore, if g(z) is univalent in U, then the followingequivalence relationship holds true (see [6]):f (z)≺ g(z)⇔ f (0) = g(0) and f (U)⊂ g(U).Entrato in redazione: 25 dicembre 2013AMS 2010 Subject Classification: 30C45.Keywords: Meromorphic functions, subordination, Hadamard product, linear operator.260 ADELA O. MOSTAFA - MOHAMED K. AOUFFor functions f (z) ∈ ∑p, given by (1) and g(z) ∈ ∑p defined byg(z) = z−p+∞∑k=1bk−pzk−p (p ∈ N), (2)the Hadamard product (or convolution) of f (z) and g(z) is given by( f ∗g)(z) = z−p+∞∑k=1ak−pbk−pzk−p = (g∗ f )(z). (3)Using the operator Qαβ ,p : ∑p→ ∑p(α ≥ 0;β >−1; p ∈ N) defined by Aqlan etal. [1], where:Qαβ ,p f (z) =z−p+Γ(α+β )Γ(β )∞∑k=1Γ(k+β )Γ(k+β+α)ak−pzk−p (α > 0)f (z) (α = 0) ,Mostafa [8] (see also [9]) defined the operator Hαp,β ,µ : Σp→ Σp as follows:For Gαβ ,p, given byGαβ ,p = z−p+Γ(α+β )Γ(β )∞∑k=1Γ(k+β )Γ(k+β +α)zk−p (4)let Gα∗β ,p,µ be defined byGαβ ,p(z)∗Gα∗β ,p,µ(z) =1zp(1− z)µ (µ > 0; p ∈ N) . (5)ThenHαp,β ,µ f (z) = Gα∗β ,p(z)∗ f (z) ( f ∈ Σp) . (6)Using (4)-(6), we haveHαp,β ,µ f (z) = z−p+Γ(β )Γ(α+β )∞∑k=1Γ(k+β +α)(µ)kΓ(k+β )(1)kak−pzk−p , (7)where f ∈ Σp is in the form (1) and (ν)n denotes the Pochhammer symbol givenby(ν)n =Γ(ν+n)Γ(ν)={1 (n = 0)ν(ν+1) . . .(ν+n−1) (n ∈ N) .It is readily verified from (7) thatz(Hαp,β ,µ f (z))′= (α+β )Hα+1p,β ,µ f (z)− (α+β + p)Hαp,β ,µ f (z) (8)ON CONVOLUTION PROPERTIES 261andz(Hαp,β ,µ f (z))′= µHαp,β ,µ+1 f (z)− (µ+ p)Hαp,β ,µ f (z) . (9)It is noticed that, putting µ = 1 in (7), we obtain the operatorHαp,β ,1 f (z) = Hαp,β f (z) = z−p+Γ(β )Γ(α+β )∞∑k=1Γ(k+α+β )Γ(k+β )ak−pzk−p, (10)andH0p,β ,1 f (z) = f (z).For −1≤ A < B≤ 1,B≥ 0 and z ∈U∗, Mogra [8] defined the classS∗p[A,B] ={f ∈∑p :−z f ′(z)f (z)≺ p1+Az1+Bz,z ∈U∗}(11)and Srivastava et al. [12] defined the class∑Kp[A,B] ={f ∈∑p :−[1+z f ′′(z)f ′(z)]≺ p1+Az1+Bz,z ∈U∗}. (12)It is clear thatf (z) ∈∑Kp[A,B]⇔−z f′(z)p∈ S∗p[A,B], (13)S∗1[2α−1,1] =∑S∗(α) (see Juneja and Reddy [5]),K1[2α−1,1] =∑K(α) (0≤ α < 1) (see Srivastava et al. [12]),S∗p[2αp−1,1] =∑S∗p(α) (0≤ α < p) (see Aouf and Hossen [2])and∑Kp[2αp −1,1] =∑Kp(α) (0≤ α < p) (see Aouf and Srivastava [3]).Using the operator Hαp,β ,µ and for−1≤B<A≤ 1,α ≥ 0,β >−1, µ > 0 andz ∈U∗ we define the classes S∗p(α,β ,µ,A,B) and Kp(α,β ,µ,A,B) as follows:S∗p(α,β ,µ,A,B) ={f ∈∑p : Hαp,β ,µ f (z) ∈∑[p,A,B],z ∈U}, (14)andKp(α,β ,µ,A,B) ={f ∈∑p : Hαp,β ,µ f (z) ∈∑Kp[A,B],z ∈U}. (15)We notice thatf (z) ∈ Kp(α,β ,µ,A,B)⇔−z f′(z)p∈ S∗p(α,β ,µ,A,B). (16)262 ADELA O. MOSTAFA - MOHAMED K. AOUF2. Main ResultsUnless otherwise mentioned, we shall assume in this paper that −1≤ A < B≤1, 0≤B< 1, α ≥ 0, β >−1, (α+β ) 6= 0, µ > 0, 0< θ < 2pi, p∈N and z∈U∗.To prove our results we need the following lemmas.Lemma 2.1 ([10]). The function f (z) defined by (1.1) is in the class ∑[p,A,B]if and only ifzp[f (z)∗ 1+(D−1)zzp(1− z)2]6= 0, (17)whereD =e−iθ +Bp(A−B) . (18)Lemma 2.2 ([10]). The function f (z) defined by (1) is in the class ∑Kp[A,B] ifand only ifzp{f (z)∗[p−{2+ p− (p−1)(D−1)}z− (p+1)(D−1)z2pzp(1− z)3]}6= 0.Lemma 2.3 ([4]). Let h be convex (univalent) in U, with ℜ[βh(z)+ γ] > 0 forall z ∈U. If p is analytic in U, with p(0) = h(0), thenp(z)+zp′(z)β p(z)+ γ≺ h(z)⇒ p(z)≺ h(z). (19)Theorem 2.4. The function f (z) defined by (1) is in the class S∗p(α,β ,µ,A,B)if and only if1+∞∑k=1[ke−iθ + pA+(k− p)Bp(A−B)]Γ(β )Γ(α+β )Γ(k+α+β )Γ(k+β )akzk 6= 0. (20)Proof. From Lemma 2.1, we find that f (z) ∈ S∗p(α,β ,µ,A,B) if and only ifzp[Hαp,β ,µ f (z)∗1+(D−1)zzp(1− z)2]6= 0.Expanding 1+(D−1)zzp(1−z)2 , we have (20) which completes the proof of Theorem 2.4.Theorem 2.5. The function f (z) defined by (1) is in the class Kp(α,β ,µ,A,B)ifand only if1−∞∑k=1[k[ke−iθ + pA+(k− p)B]p2(A−B)]Γ(β )Γ(α+β )Γ(k+α+β )Γ(k+β )akzk 6= 0. (21)ON CONVOLUTION PROPERTIES 263Proof. From Lemma 2.2, we find that f (z) ∈ Kp(α,β ,µ,A,B) if and only ifzp{Hαp,β ,µ f (z)∗[p−{2+ p− (p−1)(D−1)}z− (p+1)(D−1)z2pzp(1− z)3]}6= 0. (22)Now it can be easily shown thatz−p(1− z)−3 = z−p+∞∑k=1(k+1)(k+2)2zk−p, (23)z1−p(1− z)−3 =∞∑k=1k(k+1)2zk−p, (24)z2−p(1− z)−3 =∞∑k=1k(k−1)2zk−p. (25)Using (23)-(25) and (18) in (22), we have the desired result which completesthe proof of Theorem 2.5.Theorem 2.6. If the function f defined by (1) belongs to the classS∗p(α,β ,µ,A,B), then∞∑k=1[k+ pA+(k− p)B] Γ(β )Γ(α+β )Γ(k+α+β )Γ(k+β )|ak| ≤ p(A−B). (26)Proof. Since∣∣∣∣ke−iθ + pA+(k− p)Bp(A−B)∣∣∣∣=∣∣ke−iθ + pA+(k− p)B∣∣p(A−B) ≤k+ pA+(k− p)Bp(A−B)and ∣∣∣∣∣1+ ∞∑k=1[ke−iθ + pA+(k− p)B]p(A−B)Γ(β )Γ(α+β )Γ(k+α+β )Γ(k+β )∣∣akzk∣∣∣∣∣∣∣> 1−∞∑k=1∣∣∣∣∣[ke−iθ + pA+(k− p)B]p(A−B)∣∣∣∣∣ Γ(β )Γ(α+β ) Γ(k+α+β )Γ(k+β ) |ak|the result follows from Theorem 2.4.Using the same technique, we can also prove the following theorem.Theorem 2.7. If the function f defined by (1) belongs to the classKp(α,β ,µ,A,B), then∞∑k=1k [k+ pA+(k− p)B] Γ(β )Γ(α+β )Γ(k+α+β )Γ(k+β )|ak| ≤ p2(A−B). (27)264 ADELA O. MOSTAFA - MOHAMED K. AOUFTheorem 2.8. Let the function f (z) be defined by (1). If1+AB+(A+B)cosθ1+B2+2Bcosθ≤ α+β + p (28)and f ∈ S∗p(α+1,β ,µ,A,B) with Hαp,β ,µ f (z) 6= 0, then f ∈ S∗p(α,β ,µ,A,B).Proof. Let f (z) ∈ S∗p(α+1,β ,µ,A,B) and define the functionP(z) =−z(Hαp,β ,µ f (z))′Hαp,β ,µ f (z), (29)we see that P is analytic in U with P(0) = 1. Using the identity (8) in (29), wehaveHα+1p,β ,µ f (z)Hαp,β ,µ f (z)=− 1α+βP(z)+α+β + pα+β. (30)Differentiating (30) logarithmically and using (29), we have−z(Hα+1p,β ,µ f (z))′Hα+1p,β ,µ f (z)= p(z)+zP′(z)−P(z)+α+β + p ≺1+Az1+Bz= h(z). (31)Simple computations show that the inequalty ℜ{−h(z)+α +β + p} > 0 canbe written in the formℜ1+Az1+Bz− (α+β + p)< 0,which is equivalent to (28). Since the function h(z) is a convex function, thenapplying Lemma 2.3, we see that the subordination (31) implies P(z) ≺ h(z).This completes the proof of Theorem 2.8.Theorem 2.9. Let the function f (z) be defined by (1). If1+AB+(A+B)cosθ1+B2+2Bcosθ≤ µ+ p (32)and f ∈ S∗p(α,β ,µ+1,A,B) with Hαp,β ,µ f (z) 6= 0, then f ∈ S∗p(α,β ,µ,A,B).Proof. The proof follows in the same steps as that used in Theorem 2.8 andusing the identity (9) instead of (8).Using (16) and the fact thatHαp,β ,µ(−z f ′)(z) =−z(Hαp,β ,µ f (z))′,Theorem 2.8 yields the following theorem.ON CONVOLUTION PROPERTIES 265Theorem 2.10. Let the function f (z) be defined by (1). If (28) holds and f (z)∈Kp(α+1,β ,µ,A,B) with Hαp,β ,µ f (z) 6= 0, then f (z) ∈ Kp(α,β ,µ,A,B).REFERENCES[1] E. Aqlan - J. M. Jahangiri - S. R. Kulkarni, Certain integral operators applied top-valent functions, J. Nat. Geom. 24 (2005), 111–120.[2] M. K. Aouf - H. M. Hossen, New criteria for meromorphic p-valent starlike func-tions, Tsukuba J. Math. 17 (2) (1993), 481–486.[3] M. K. Aouf - H. M. Srivastava, A new criteria for meromorphically p-valent con-vex functions of order alpha, Math. Sci. Res. Hot-line 1 (8) (1997), 7–12.[4] P. J. Eenigenburg - S. S. Miller - P. T. Mocanu - M. O. Reade, Second order differ-ential inequalities in the complex plane, J. Math. Anal. Appl. 6 (1978), 289–305.[5] O. P. Juneja - T. R. Reddy, Meromorphic starlike univalent functions with positivecoefficients, Ann. Univ. Mariae Curie-Sklod. 39 (9) (1985), 65–75.[6] S. S. Miller - P. T. Mocanu, Differential Subordination: Theory and Applications,Series on Monographs and Textbooks in Pure and Applied Mathematics 225, Mar-cel Dekker Inc., New York and Basel, 2000.[7] M. L. Mogra, Meromorphic multivalent functions with positive coefficients I,Math. Japon. 35 (1990), 1–11.[8] A. O. Mostafa, Inclusion results for certain subclasses of p-valent meromorphicfunctions associated with a new operator, J. Inequal. Appl. (2012), 1–11.[9] A. O. Mostafa - M. K. Aouf, Properties of p-valent meromorphic functions asso-ciated with linear operator, Electronic J. Math. Anal. Appl. 2 (4) (2013), 1–5.[10] N. Sarkar - P. Goswami - and M. K. Aouf, Convolution properties for certainclasses of meromorphic p-valent functions defined by subordination, Thai J. Math.10 (3) (2012), 577–585.[11] H. M. Srivastava - H. M. Hossen M. K. Aouf, A unified presentation of someclasses of meromorphically multivalent functions, Comput. Math. Appl. 38(1999), 63–70.ADELA O. MOSTAFADepartment of Mathematics, Faculty of ScienceMansoura University, Mansoura 35516, Egypte-mail: adelaeg254@yahoo.com266 ADELA O. MOSTAFA - MOHAMED K. AOUFMOHAMED K. AOUFDepartment of Mathematics, Faculty of ScienceMansoura University, Mansoura 35516, Egypte-mail: mkaouf127@yahoo.com

On convolution properties for certain classes of p-valent meromorphic functions defined by linear operator

In this paper, by making use of convolution, we obtain some interesting results for certain family of meromorphic p-valent functions defined by new linear operator.    </p

Adela Mostafa

M. K. Aouf

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On convolution properties for certain classes of p-valent meromorphic functions defined by linear operator

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