MATEMATIQKI VESNIK 65, 1 (2013), 14 28 March 2013 originalni nauqni rad research paper CERTAIN SUBCLASSES OF UNIFORMLY STARLIKE AND CONVEX FUNCTIONS DEFINED BY CONVOLUTION WITH NEGATIVE COEFFICIENTS M.K. Aouf, A.A. Shamandy, A.O. Mostafa and A.K. Wagdy Abstract. The aim of this paper is to obtain coefficient estimates, distortion theorems, convex linear combinations and radii of close-to-convexity, starlikeness and convexity for functions belonging to the class T S(g, λ; α, β). Furthermore partial sums f n() of functions f() in the class T S(g, λ; α, β) are considered and sharp lower bounds for the ratios of real part of f() to f n () and f () to f n () are determined. 1. Introduction Let A denote the class of functions of the form f() = + a k k, (1.1) which are analytic in the open unit disc U = { C : < 1}. Let g A be given by g() = + b k k, (1.2) the Hadamard product (or convolution) f g of f and g is defined (as usual) by (f g)() = + a k c k k = (g f)(). (1.3) Following Goodman ([6] and [7]), Ronning ([11 and [12]) introduced and studied the following subclasses: (i) A function f() of the form (1.1) is said to be in the class S p (α, β) of β-uniformly starlike functions if it satisfies the condition: { f } () Re f () α > β f () f () 1 ( U), (1.4) where 1 α < 1 and β 0. 2010 AMS Subject Classification: 30C45. Keywords and phrases: Analytic function; Hadamard product; distortion theorems; partial sums. 14
Subclasses of uniformly starlike and convex functions 15 (ii) A function f() of the form (1.1) is said to be in the class UCV (α, β) of β-uniformly convex functions if it satisfies the condition: { Re 1 + f } () f () α > β f () f () ( U), (1.5) where 1 α < 1 and β 0. It follows from (1.4) and (1.5) that f() UCV (α, β) f () S p (α, β). (1.6) For 1 α < 1, 0 λ 1 and β 0, we let S(g, λ; α, β) be the subclass of A consisting of functions f() of the form (1.1) and functions g() of the form (1.2) and satisfying the analytic criterion: { } (f g) Re () (1 λ)(f g)()+λ (f g) () α (f g) > β () (1 λ)(f g)()+λ(f g) () 1. (1.7) (1 ) Remark 1. (i) Putting g() = in the class S(g, λ; α, β), we obtain the class S p (λ, α, β) defined by Murugusundaramoorthy and Magesh [10]. (1 ) 2 (ii) Putting g() = in the class S(g, λ; α, β), we obtain the class U CV (λ, α, β) defined by Murugusundaramoorthy and Magesh [10]. Let T denote the subclass of A consisting of functions of the form: f() = a k k (a k 0). (1.8) Further, we define the class T S(g, λ; α, β) by We note that: T S(g, λ; α, β) = S(g, λ; α, β) T (1.9) (i) T S( (1 ), 0; α, 1) = T S p(α) and T S( (1 ), 0; α, 1) = UCT (α) (see Bharati 2 et al. [2]); (ii) T S( (1 ), 0; α, β) = T S p(α, β) and T S( (1 ), 0; α, β) = UCT (α, β) (see 2 Bharati et al. [2]); (iii) T S( (1 ), 0; α, 0) = T (α) and T S( (1 ), 0; α, 0) = C (α) (see Silverman 2 [15]); (iv) T S( + (a) k 1 (c) k 1 k, 0; α, β) = T S(α, β) (c 0, 1, 2,... ) (see Murugusundaramoorthy and Magesh [8, 9]); (v) T S( + kn k, 0; α, β) = T S(n, α, β) (n N 0 = N {0}, where N = {1, 2,... }) (see Rosy and Murugusundaramoorthy [13]); (vi) T S( (1 ), λ; α, β) = T S p(λ, α, β) and T S( (1 ), λ; α, β) = UCT (λ, α, β) 2 (see Murugusundaramoorthy and Magesh [10]); (vii) T S( + ( k + δ 1 ) δ k, 0; α, β) = D(β, α, δ) (δ > 1) (see Shams et al. [14]);
16 M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy (viii) T S( + [1 + δ (k 1)]n k, 0; α, β) = T S δ (n, α, β) (δ 0, n N 0 ) (see Aouf and Mostafa [1]). Also we note that: (i) T S( + Γ k (α 1 ) k, λ; α, β) = T S q,s (α 1 ; λ, α, β) { } (H = {f T : Re q,s(α 1,β 1)f()) (1 λ)h q,s (α 1,β 1 )f ()+λ(h q,s (α 1,β 1 )f()) α (H > β q,s (α 1,β 1 )f()) }, 1 (1 λ)h q,s(α 1,β 1)f()+λ (H q,s(α 1,β 1)f()) where 1 α < 1, 0 λ 1, β 0, U and Γ k (α 1 ) is defined by Γ k (α 1 ) = (α 1 ) k 1 (α q ) k 1 (β 1 ) k 1 (β s ) k 1 (1) k 1 (1.10) (α i > 0, i = 1,..., q; β j > 0, j = 1,..., s; q s + 1, q, s N 0 ), where the operator H q,s (α 1, β 1 ) was introduced and studied by Diok and Srivastava (see [4] and [5]), which is a generaliation of many other linear operators considered earlier; (ii) T S( + [ ] m l+1+µ(k 1) l+1 k, λ; α, β) = T S(m, µ, l, λ; α, β) { { } (I = f T : Re m (µ,l)f()) (1 λ)i m (µ,l)f()+λ(i m (µ,l)f()) α (I > β m (µ,l)f()) }, (1 λ)i m (µ,l)f()+λ(i m (µ,l)f()) 1 where 1 α < 1, 0 λ 1, β 0, m N 0, µ, l 0, U and the operator I m (µ, l) was defined by Cătaş et al. (see [3]), which is a generaliation of many other linear operators considered earlier; (iii) T S( + C k(b, µ) k, λ; α, β) = T S(b, µ, λ; α, β) = { { } } (J f T : Re µ b f()) α (J > β µ b (1 λ)j µ b f ()+λ(j µ b f f()) 1, ()) (1 λ)j µ b f ()+λ(j µ b f ()) where 1 α < 1, 0 λ 1, β 0, U and C k (b, µ) is defined by ( ) µ 1 + b C k (b, µ) = (b C \ Z 0 k + b }, Z 0 = Z \ N), (1.11) where the operator J µ b was introduced by Srivastava and Attiya (see [18]), which is a generaliation of many other linear operators considered earlier. 2. Coefficient estimates Unless otherwise mentioned, we shall assume in the reminder of this paper that, 1 α < 1, 0 λ 1, β 0 and U. Theorem 1. A function f() of the form (1.1) is in the class S(g, λ; α, β) if {k (1 + β) (α + β) [1 + λ (k 1)]} b k a k, (2.1) where b k+1 b k > 0 (k 2).
Subclasses of uniformly starlike and convex functions 17 Proof. Assume that the inequality (2.1) holds true. Then we have β (f g) () (1 λ) (f g)() + λ(f g) () 1 { (f g) } () Re (1 λ) (f g)() + λ(f g) () 1 (1 + β) (f g) () (1 λ) (f g)() + λ(f g) () 1 (1 + β) (1 λ) (k 1) b k a k k 1 1 [1 + λ (k 1)] b k a k k 1. This completes the proof of Theorem 1. Theorem 2. A necessary and sufficient condition for the function f() of the form (1.8) to be in the class T S(g, λ; α, β) is that {k (1 + β) (α + β) [1 + λ (k 1)]} a k b k. (2.2) Proof. In view of Theorem 1, we need only to prove the necessity. If f() T S(g, λ; α, β) and is real, then 1 k a k b k k 1 (1 λ) (k 1) a k b k k 1 1 α β [1 + λ (k 1)] a k b k k 1 1. [1 + λ (k λ)] a k b k k 1 Letting 1 along the real axis, we obtain {k (1 + β) (α + β) [1 + λ (k 1)]} a k b k. This completes the proof of Theorem 2. Corollary 1. T S(g, λ; α, β). Then a k The result is sharp for the function f() = Let the function f() defined by (1.8) be in the class {k (1 + β) (α + β) [1 + λ (k 1)]} b k (k 2). (2.3) {k (1 + β) (α + β) [1 + λ (k 1)]} b k k (k 2). (2.4) By taking b k = Γ k (α 1 ), where Γ k (α 1 ) is defined by (1.10), in Theorem 2, we have:
18 M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy Corollary 2. A necessary and sufficient condition for the function f() of the form (1.8) to be in the class T S q,s (α 1 ; λ, α, β) is that {k (1 + β) (α + β) [1 + λ (k 1)]} Γ k (α 1 )a k. By taking b k = [ ] m l+1+µ(k 1) l+1 (m N0, µ, l 0), in Theorem 2, we have: Corollary 3. A necessary and sufficient condition for the function f() of the form (1.8) to be in the class T S(m, µ, l, λ; α, β) is that [ ] m {k (1 + β) (α + β) [1 + λ (k 1)]} l+1+µ(k 1) l+1 ak. By taking b k = C k (b, µ), where C k (b, µ) defined by (1.11), in Theorem 2, we have: Corollary 4. A necessary and sufficient condition for the function f() of the form (1.8) to be in the class T S(b, µ, λ; α, β) is that {k (1 + β) (α + β) [1 + λ (k 1)]} C k (b, µ) a k. 3. Distortion theorem Theorem 3. Let the function f() of the form (1.8) be in the class T S(g, λ; α, β). Then for = r < 1, we have and f() r f() r + r 2 (3.1) r 2, (3.2) provided that b k+1 b k > 0 (k 2). The equalities in (3.1) and (3.2) are attained for the function f() given by f() = 2, (3.3) at = r and = re i(2k+1)π (k Z). Proof. Since for k 2, {k (1 + β) (α + β) [1 + λ (k 1)]} b k,
using Theorem 2, we have Subclasses of uniformly starlike and convex functions 19 that is, that a k {k (1 + β) (α + β) [1 + λ (k 1)]} a k b k, (3.4) a k. (3.5) From (1.8) and (3.5), we have and f() r r 2 f() r + r 2 a k r a k r + This completes the proof of Theorem 3. r 2 r 2. Theorem 4. Let the function f() of the form (1.8) be in the class T S(g, λ; α, β). Then for = r < 1, we have f () r 2 () r (3.6) and f 2 () () r + r, (3.7) provided that b k+1 b k > 0 (k 2). The result is sharp for the function f() given by (3.3). Proof. From Theorem 2 and (3.5), we have 2 () ka k, and the remaining part of the proof is similar to the proof of Theorem 3. 4. Convex linear combinations Theorem 5. Let µ υ 0 for υ = 1, 2,..., l and l υ=1 µ υ 1. If the functions F υ () defined by F υ () = a k,υ k (a k,υ 0; υ = 1, 2,..., l), (4.1) are in the class T S(g, λ; α, β) for every υ = 1, 2,..., l, then the function f() defined by ( f() = l ) µ υ a k,υ k υ=1 is in the class T S(g, λ; α, β).
20 M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy Proof. Since F υ () T S(g, λ; α, β), it follows from Theorem 2 that {k (1 + β) (α + β) [1 + λ (k 1)]} a k,υ b k, (4.2) for every υ = 1, 2,..., l. Hence ( l ) {k (1 + β) (α + β) [1 + λ (k 1)]} µ υ a k,υ b k υ=1 ( = l ) µ υ {k (1 + β) (α + β) [1 + λ (k 1)]} a k,υ b k υ=1 () l µ υ. υ=1 By Theorem 2, it follows that f() T S(g, λ; α, β). Corollary 5. The class T S(g, λ; α, β) is closed under convex linear combinations. Theorem 6. Let f 1 () = and f k () = {k (1 + β) (α + β) [1 + λ (k 1)]} b k k (k 2). (4.3) Then f() is in the class T S(g, λ; α, β) if and only if it can be expressed in the form: f() = µ k f k (), (4.4) where µ k 0 and k=1 µ k = 1. Proof. Assume that f() = µ k f k () = k=1 k=1 {k (1 + β) (α + β) [1 + λ (k 1)]} b k µ k k. (4.5) Then it follows that {k(1+β) (α+β)[1+λ(k 1)]}b k 1 α 1 α {k(1+β) (α+β)[1+λ(k 1)]}b k µ k So, by Theorem 2, f() T S(g, λ; α, β). = µ k = 1 µ 1 1. (4.6) Conversely, assume that the function f() defined by (1.8) belongs to the class T S(g, λ; α, β). Then a k {k (1 + β) (α + β) [1 + λ (k 1)]} b k (k 2). (4.7)
Setting and Subclasses of uniformly starlike and convex functions 21 µ k = {k (1 + β) (α + β) [1 + λ (k 1)]} a kb k (k 2), (4.8) µ 1 = 1 µ k, (4.9) we can see that f() can be expressed in the form (4.4). This completes the proof of Theorem 6. Corollary 6. The extreme points of the class T S(g, λ; α, β) are the functions f 1 () = and f k () = {k (1 + β) (α + β) [1 + λ (k 1)]} b k k (k 2). (4.10) 5. Radii of close-to-convexity, starlikeness and convexity Theorem 7. Let the function f() defined by (1.8) be in the class T S(g, λ; α, β). Then f() is close-to-convex of order ρ (0 ρ < 1) in < r 1, where r 1 = inf k 2 { (1 ρ) {k (1 + β) (α + β) [1 + λ (k 1)]} bk k () The result is sharp, with the extremal function f() given by (2.4). Proof. We must show that f () 1 1 ρ for < r 1, where r 1 is given by (5.1). Indeed we find from (1.8) that Thus f () 1 1 ρ, if f () 1 ka k k 1. } 1 k 1. (5.1) ( ) k a k k 1 1. (5.2) 1 ρ But, by Theorem 2, (5.2) will be true if ( ) k k 1 {k (1 + β) (α + β) [1 + λ (k 1)]} b k, 1 ρ that is, if { (1 ρ) {k (1 + β) (α + β) [1 + λ (k 1)]} bk k () Theorem 7 follows easily from (5.3). } 1 k 1 (k 2). (5.3)
22 M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy Theorem 8. Let the function f() defined by (1.8) be in the class T S(g, λ; α, β). Then f() is starlike of order ρ (0 ρ < 1) in < r 2, where r 2 = inf k 2 { (1 ρ) {k (1 + β) (α + β) [1 + λ (k 1)]} bk (k ρ) () The result is sharp, with the extremal function f() given by (2.4). Proof. We must show that f () f () 1 1 ρ for < r 2, where r 2 is given by (5.4). Indeed we find from (1.8) that Thus f () f() 1 1 ρ, if f () f () 1 (k 1) a k k 1 1. a k k 1 } 1 k 1. (5.4) (k ρ) a k k 1 1 ρ 1. (5.5) But, by Theorem 2, (5.5) will be true if (k ρ) k 1 1 ρ {k (1 + β) (α + β) [1 + λ (k 1)]} b k, that is, if { (1 ρ) {k (1 + β) (α + β) [1 + λ (k 1)]} bk (k ρ) () } 1 k 1 (k 2). (5.6) Theorem 8 follows easily from (5.6). Corollary 7. Let the function f() defined by (1.8) be in the class T S(g, λ; α, β). Then f() is convex of order ρ (0 ρ < 1) in < r 3, where r 3 = inf k 2 { (1 ρ) {k (1 + β) (α + β) [1 + λ (k 1)]} bk k (k ρ) () The result is sharp, with the extremal function f() given by (2.4). } 1 k 1. (5.7)
Subclasses of uniformly starlike and convex functions 23 6. A family of integral operators In view of Theorem 2, we see that d k k is in the class T S(g, λ; α, β) as long as 0 d k a k for all k. In particular, we have: Theorem 9. Let the function f() defined by (1.8) be in the class T S(g, λ; α, β) and c be a real number such that c > 1. Then the function F () defined by F () = c + 1 c t c 1 f(t) dt (c > 1), (6.1) also belongs to the class T S(g, λ; α, β). where Proof. From the representation (6.1) of F (), it follows that d k = 0 F () = d k k, ( ) c + 1 a k a k (k 2). c + k On the other hand, the converse is not true. This leads to a radius of univalence result. Theorem 10. Let the function F () = a k k (a k 0) be in the class T S(g, λ; α, β), and let c be a real number such that c > 1. Then the function f() given by (6.1) is univalent in < R, where R = inf k 2 The result is sharp. Proof. From (6.1), we have { (c + 1) {k (1 + β) (α + β) [1 + λ (k 1)]} bk f() = 1 c c F () c + 1 k (c + k) () = ( ) c + k a k k. c + 1 In order to obtain the required result, it suffices to show that where R is given by (6.2). Now Thus f () 1 < 1 if f () 1 < 1 wherever < R, f () 1 k (c + k) (c + 1) a k k 1. } 1 k 1. (6.2) k (c + k) (c + 1) a k k 1 < 1. (6.3)
24 M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy But Theorem 2 confirms that {k (1 + β) (α + β) [1 + λ (k 1)]} a k b k Hence (6.3) will be satisfied if 1. (6.4) that is, if k (c + k) (c + 1) k 1 < {k (1 + β) (α + β) [1 + λ (k 1)]} b k, { (c + 1) {k (1 + β) (α + β) [1 + λ (k 1)]} bk < k (c + k) () } 1 k 1 (k 2). (6.5) Therefore, the function f() given by (6.1) is univalent in < R. Sharpness of the result follows if we take f() = (c + k) () (c + 1) {k (1 + β) (α + β) [1 + λ (k 1)]} b k k (k 2). (6.6) 7. Partial sums Following the earlier works by Silverman [16] and Siliva [17] on partial sums of analytic functions, we consider in this section partial sums of functions in the class T S(g, λ; α, β) and obtain sharp lower bounds for the ratios of real part of f() to f n () and f () to f n (). Theorem 11. Define the partial sums f 1 () and f n () by f 1 () = and f n () = + n a k k, (n N \ {1}). Let f() T S(g, λ; α, β) be given by (1.8) and satisfy condition (2.2) and { 1, k = 2, 3,..., n, c k, k = n + 1, n + 2,..., where, for convenience, (7.1) Then and c k = {k (1 + β) (α + β) [1 + λ (k 1)]} b k. (7.2) { } f() Re > 1 1 ( U; n N), (7.3) f n () Re { } fn () >. (7.4) f() 1 +
Subclasses of uniformly starlike and convex functions 25 Proof. For the coefficients c k given by (7.2) it is not difficult to verify that Therefore we have By setting n a k + c k+1 > c k > 1. (7.5) a k c k a k 1. (7.6) { ( f() g 1 () = f n () 1 1 )} = 1 + and applying (7.6), we find that Now g1() 1 g 1 ()+1 1 if c g 1 () 1 g 1 () + 1 2 2 n a k n a k + From condition (2.2), it is sufficient to show that which is equivalent to n a k + n (c k 1) a k + a k k 1 1 + n, (7.7) a k k 1 a k a k 1. a k c k a k a k. (7.8) (c k ) a k 0 (7.9) which readily yields the assertion (7.3) of Theorem 11. In order to see that gives sharp result, we observe that for = r e iπ n as 1. Similarly, if we take { fn () g 2 () = (1 + ) f() c } = 1 1 + f() = + (7.10) that f() f n() = 1 + n 1 1 (1 + ) a k k 1 1 +, (7.11) a k k 1
26 M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy and making use of (7.6), we can deduce that (1 + c g 2 () 1 ) g 2 () + 1 2 2 n a k (1 ) a k which leads us immediately to the assertion (7.4) of Theorem 11. a k (7.12) The bound in (7.4) is sharp for each n N with the extremal function f() given by (7.10). The proof of Theorem 11 is thus completed. and Theorem 12. If f() of the form (1.8) satisfies condition (2.2), then { f } () Re f n 1 n + 1, (7.13) () Re { } f n () f (7.14) () n + 1 + where c k is defined by (7.2) and satisfies the condition { k, k = 2, 3,..., n, c k k, k = n + 1, n + 2,... (7.15) The results are sharp with the function f() given by (7.10). Proof. By setting we obtain Now g() 1 g()+1 1 if g() = n + 1 = 1 + = 1 + { f ( () f n () 1 n + 1 1 + )} ka k k 1 + n ka k k 1 1 + n, ka k k 1 ka k k 1 1 + n, (7.16) ka k k 1 g() 1 g() + 1 2 2 n n ka k + n + 1 ka k ka k n + 1 ka k. (7.17) ka k 1, (7.18)
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28 M.K. Aouf, A.A. Shamandy, A.O. Mostafa, A.K. Wagdy [13] T. Rosy, G. Murugusundaramoorthy, Fractional calculus and their applications to certain subclass of uniformly convex functions, Far East J. Math. Sci. 15 (2004), 231 242. [14] S. Shams, S.R. Kulkarni, J.M. Jahangiri, Classes of uniformly starlike and convex functions, Internat. J. Math. Math. Sci. 55 (2004), 2959 2961. [15] H. Silverman, Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51 (1975), 109 116. [16] H. Silverman, Partial sums of starlike and convex functions J. Math. Anal. Appl. 209 (1997), 221 227. [17] E.M. Silvia, On partial sums of convex functions of order α, Houston J. Math. 11 (1985), 397 404. [18] H.M. Srivastava, A.A. Attiya, An integral operator associated with the Hurwit-Lerch Zeta function and differential subordination, Integral Transform. Spec. Funct. 18 (2007), 207 216. (received 13.03.2011; in revised form 01.04.2012; available online 01.05.2012) Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt E-mail: mkaouf127@yahoo.com, aashamandy@hotmail.com, adelaeg254@yahoo.com, awagdyfos@yahoo.com