International Journal of Analysis and Applications ISSN 2291-8639 Volume 11, Number 2 (2016), 70-80 http://www.etamaths.com ALPHA CONVEX FUNCTIONS ASSOCIATED WITH CONIC DOMAINS KHALIDA INAYAT NOOR1, NASIR KHAN1 AND KRZYSZTOF PIEJKO2,∗ Abstract. In this paper we define a new class k − UMα [A,B] of Janowski type k−uniformly alpha convex functions. We use the method of differential subordinations theory to obtain some new results like sufficient condition, inclusion relations, coefficient estimate and covering properties. The results presented here include a number of well-known results as their special cases. 1. Introduction Let A denote the class of functions f (z) of the form (1.1) f (z) = z + ∞∑ n=2 anz n, which are analytic in the unit disk E = {z ∈ C : |z| < 1}. Furthermore S represents class of all functions in A which are univalent in E. For two functions f(z) and g(z) analytic in A, we say that f(z) is subordinate to g(z) in E (and write f ≺ g or f(z) ≺ g(z)), if there exists an analytic function w(z) such that |w(z)| ≤ |z| and f(z) = g (w(z)) for z ∈ E. If g(z) is univalent in E then f(z) ≺ g(z) if and only if f (0) = g (0) and f (E) ⊂ g (E). The idea of subordination goes back to Lindelöf [9]. Subordination was more formally introduced and studied by Littelwood [10] and later by Rogosinski [20] and [19]. The concept of subordination was considered by Miller [12] and further investigated by Noor et al. [16] and many others see [9],[21]. Definition 1. A function p (z) is said to be in the class P [A,B] , if it is analytic in E with p (0) = 1 and p (z) ≺ 1 + Az 1 + Bz , − 1 ≤ B < A ≤ 1. This class was presented by Janowski [3] and explored by a few creators. Kanas and Wísniowska [4],[5] presented and examined the class k−ST of k−starlike functions and the relating class k−UCV of k−uniformly convex functions. There were characterized subject to the conic region Ωk, k ≥ 0, as Ωk = { u + iv : u > k √ (u− 1)2 + v2 } . This domain represents the right half plane, a parabola, a hyperbola and an ellipse for k = 0,k = 1, 0 < k < 1 and k > 1 respectively. The extremal functions for these conic regions are (1.2) pk (z) =   1+z 1−z , k = 0, 1 + 2 π2 ( log 1+ √ z 1− √ z )2 , k = 1, 1 + 2 1−k2 sinh 2 {( 2 π arccos k ) arctan h √ z } , 0 < k < 1, 1 + 2 k2−1 sin ( π 2R(t) ∫ u(z)√ t 0 dx√ 1−x2 √ 1−(tx)2 ) + 1 k2−1, k > 1, where u(z) = z − √ t 1 − √ tx , (z ∈ E) , 2010 Mathematics Subject Classification. 30C45, 30C50. Key words and phrases. subordination; Janowski functions; conic region. c©2016 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 70 ALPHA CONVEX FUNCTIONS 71 and t ∈ (0, 1) and z is chosen such that k = cosh ( πR′(t) 4R(t) ) . Here R(t) is Legendre’s complete elliptic integral of first kind and R′(t) is the complementary integral of R (t). If pk (z) = 1 + δkz + · · · , then it is shown in [5] that from (1.2), one can have (1.3) δk =   8(arccos k)2 π2(1−k2) , 0 ≤ k < 1, 8 π2 , k = 1, π2 4(k2−1) √ t(1+t)R2(t) , k > 1. Using the concepts of Janowski functions and the conic regions, Noor et al. [16] gave the following Definition 2. [16]A function p (z) is said to be in the class k −P [A,B] , if and only if p (z) ≺ (A + 1) pk (z) − (A− 1) (B + 1) pk (z) − (B − 1) , k ≥ 0, where pk (z) is defined in (1.2) and −1 ≤ B < A ≤ 1. Geometrically, the function p (z) ∈ k − P [A,B], takes all values from the domain Ωk[A,B], −1 ≤ B < A ≤ 1,k ≥ 0 which is defined as Ωk[A,B] = { w : Re ( (B − 1) w (z) − (A− 1) (B + 1) w (z) − (A + 1) ) > k ∣∣∣∣(B − 1) w (z) − (A− 1)(B + 1) w (z) − (A + 1) − 1 ∣∣∣∣ } . The domain Ωk[A,B] retains the conic domain Ωk inside the circular region defined by Ω[A,B]. The impact of Ω[A,B], on the conic domain Ωk, changes the original shape of the conic regions. The ends of hyperbola and parabola get closer to one another but never meet anywhere and the ellipse gets the oval shape. When A → 1,B →−1 the radius of the circular disk defined by Ω[A,B] tends to infinity, consequently the arm of the hyperbola and parabola expands to the oval terns into ellipse. We see that Ωk[1,−1] = Ωk, the conic domain defined by Kanas and Wísniowska [4]. Now using Janowski functions and the conic regions, we give the following Definition 3. A function f (z) ∈ A is said to be in the class k−UMα [A,B] , k ≥ 0, 0 ≤ α ≤ 1,−1 ≤ B < A ≤ 1, if and only if (1.4) J (α,f; z) ∈ k −P [A,B] , where J (α,f; z) = (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) . Special Cases: (i) k −UM0 [A,B] = k −ST [A,B], k −UM1 [A,B] = k −UCV [A,B] , the classes introduced by Noor et al. in [16]. (ii) k −UM0 [1,−1] = k −ST and k −UM1 [1,−1] = k −UCV, we get the classes investigated by Kanas and Wisniowska [4], [5]. (iii) k −UMα [1,−1] = k −UMα, we have the class introduced and studied by Kanas [7]. (iv) 0 −UM0 [A,B] = S [A,B] and 0 −UM1 [A,B] = C [A,B] , the well-known classes of Janowski starlike and Janowski convex functions, respectively, introduced by Janowski [3]. Definition 4. Let SS∗ (β) denote the class of strongly starlike functions of order β, SS∗ (β) = { f ∈ A : ∣∣∣∣arg zf′(z)f (z) ∣∣∣∣ < βπ2 z ∈ E } , β ∈ (0, 1) , which was introduced in [24] and [1]. In this paper, several interesting subordination results are derived which yield sufficient condition, inclusion relations, coefficient estimate, covering result and order of strongly starlikeness in the class of uniformly alpha convex function. To avoid repetitions, it is admitted once that 0 ≤ α ≤ 1,k ≥ 0, and −1 ≤ B < A ≤ 1. 72 NOOR, KHAN AND PIEJKO 2. Preliminary results To prove our main results we need the following Lemmas. Lemma 1. [19]Let f(z) be subordinate to g(z), with f(z) = 1 + ∞∑ n=1 anz n, g(z) = 1 + ∞∑ n=1 bnz n. If g(z) is univalent in E and g(E) is convex, then |an| ≤ |b1|. Lemma 2. [11]Let F be analytic and convex in E. If f,g ∈ A and f,g ≺ F, then for t ∈ [0, 1] (1 − t)f + tg ≺ F. Lemma 3. [14] Let k ≥ 0 and let δ,σ be any complex numbers with δ 6= 0 and < (( 2k+1−A 2k+1−B ) δ + σ ) > 0. If p(z) is analytic in E and p(0) = 1 and satisfies (2.1a) p(z) + zp ′ (z) δp(z) + σ ≺ pk(A,B; z), where pk(A,B; z) = (A + 1) pk (z) − (A− 1) (B + 1) pk (z) − (B − 1) , and q(z) is an analytic solution of q(z) + zq(z) δq(z) + σ = pk(A,B; z) then function q(z) is univalent p(z) ≺ q(z) ≺ pk(A,B; z) and q(z) is the best dominant of (2.1a) and is given as q(z) = [ δ ∫ 1 0 ( tδ+σ−1 exp ∫ tz t pk(A,B,z) − 1 u du )δ dt ]−1 − σ δ . Lemma 4. [18] Let a function p (z) be analytic in E and has the form p(z) = 1 + ∞∑ n=m cnz n , cm 6= 0, with p (z) 6= 0 for |z| < 1. If there exists a point z◦, |z◦| < 1, such that |arg p(z)| < π 2 θ for |z| < |z◦| and |arg p(z◦)| = π 2 θ, for some θ > 0, then we have z◦p ′(z◦) p(z◦) = ilθ, where l ≥ m 2 (x + 1 x ) ≥ m when arg{p(z◦)} = π 2 θ and l ≤− m 2 (x + 1 x ) ≤−m when arg{p(z◦)} = − π 2 θ, where (p (z◦)) 1 θ = ±ix and x > 0. ALPHA CONVEX FUNCTIONS 73 3. Main results Theorem 1. A function f (z) ∈ k −UMα[A,B], if it satisfies the condition ∞∑ n=1 zn (k,α,A,B) < |B −A| , where (3.1) zn (k,α,A,B) = ∑∞ n=2[|{2 (k + 1) (1 −n) (1 −α (1 −n)) + ((A + 1) (n + 1) −(B + 1) ( 2n + α ( 1 −n2 )) )}| |an| + ∑n−1 j=2 |{2 (k + 1) ((1 − j) −α (n + 1 − 2j)) + ((A + 1) − (B + 1) ((1 − 2α) j + α (1 + n)))}(n + 1 − j) | |ajan+1−j|]. Proof. Assume that (3.1) holds, then it suffices to show that (3.2) k ∣∣∣∣(B − 1)J (α,f; z) − (A− 1)(B + 1)J (α,f; z) − (A + 1) − 1 ∣∣∣∣−Re [ (B − 1)J (α,f; z) − (A− 1) (B + 1)J (α,f; z) − (A + 1) − 1 ] < 1. We have k ∣∣∣∣(B − 1)J (α,f; z) − (A− 1)(B + 1)J (α,f; z) − (A + 1) − 1 ∣∣∣∣−Re [ (B − 1)J (α,f; z) − (A− 1) (B + 1)J (α,f; z) − (A + 1) − 1 ] ≤ (k + 1) ∣∣∣∣∣ (B − 1) ( (1 −α) zf′(z)f′(z) + αf (z) (zf′(z))′ ) − (A− 1)f (z) f′(z) (B + 1) ( (1 −α) zf′(z)f′ (z) + αf (z) (zf′(z))′ ) − (A + 1)f (z) f′(z) − 1 ∣∣∣∣∣ = 2 (k + 1) ∣∣∣∣∣ (1 −α) f (z) f ′(z) − (1 −α) zf′(z)f′(z) −αzf (z) f′′(z) (B + 1) ( (1 −α) zf′(z)f′(z) + αf (z) (zf′(z))′ ) − (A + 1)f (z) f′(z) ∣∣∣∣∣ . (3.3) Now we have zf′(z)f′(z) = z ( ∞∑ n=0 nanz n−1 )( ∞∑ n=0 nanz n−1 ) (3.4) = 1 z ( ∞∑ n=0 nanz n )( ∞∑ n=0 nanz n ) = 1 z ∞∑ n=0   n∑ j=0 j (n− j) ajan−j  zn = ∞∑ n=0   n∑ j=0 j (n− j) ajan−j  zn−1 = z + ∞∑ n=3   n∑ j=0 j (n− j) ajan−j  zn−1 = z + ∞∑ n=2  n+1∑ j=0 j (n + 1 − j) ajan+1−j  zn = z + ∞∑ n=2  2nan + n−1∑ j=2 j (n + 1 − j) ajan+1−j  zn.(3.5) 74 NOOR, KHAN AND PIEJKO Proceeding on the same way we have (3.6) f (z) f′(z) = z + ∞∑ n=2  (n + 1) an + n−1∑ j=2 (n + 1 − j) ajan+1−j  zn and (3.7) zf (z) f′′(z) = ∞∑ n=2  n (n− 1) an + n−1∑ j=2 (n + 1 − j) (n− j) ajan+1−j  zn. Using the equalities (3.5),(3.6) and (3.7) in (3.3), the equation (3.3) in simplified form can be written as k ∣∣∣∣(B − 1)J (α,f; z) − (A− 1)(B + 1)J (α,f; z) − (A + 1) − 1 ∣∣∣∣−Re [ (B − 1)J (α,f; z) − (A− 1) (B + 1)J (α,f; z) − (A + 1) − 1 ] ≤ 2 (k + 1) ∣∣∣∣∣ (1 −α) f (z) f ′(z) − (1 −α) zf′(z)f′ (z) −αzf (z) f′′(z) (B + 1) ( (1 −α) zf′(z)f′(z) + αf (z) (zf′(z))′ ) − (A + 1)f (z) f′(z) ∣∣∣∣∣ ≤ 2 (k + 1)   ∑∞ n=2[|(1 −n) (1 −α (1 −n))| |an| + ∑n−1 j=2 |[(1 − j) −α (n + 1 − 2j)] (n + 1 − j)| |ajan+1−j|]     |B −A|− ∑∞ n=2[ ∣∣((A + 1) (n + 1) − (B + 1) (2n + α(1 −n2)))∣∣ |an| − ∑n−1 j=2 |((A + 1) − (B + 1) ((1 − 2α) j + α (1 + n))) (n + 1 − j)| |ajan+1−j|]   . The last expression is bounded by 1, if∑∞ n=2[|{2 (k + 1) (1 −n) (1 −α (1 −n)) + ((A + 1) (n + 1) −(B + 1) ( 2n + α ( 1 −n2 )) )}| |an| + ∑n−1 j=2 |{2 (k + 1) ((1 − j) −α (n + 1 − 2j)) + ((A + 1) − (B + 1) ((1 − 2α) j + α (1 + n)))}(n + 1 − j) | |ajan+1−j|] < |B −A| . This completes the proof. � Putting α = 0, in Theorem 1, we have the result below which is comparable to the one obtained by Noor and Malik [15]. Corollary 1. A function f ∈ k −ST [A,B] , if it satisfies the condition ∞∑ n=2 {2 (k + 1) (n− 1) + |n (B + 1) + (A + 1)|} |an| < |B −A| . Putting α = 0, A = 1 and B = −1 in Theorem 1, we can obtain the following result which improves the result of Kanas and Wísniowska [4]. Corollary 2. A function f ∈ k −ST, if it satisfies the condition ∞∑ n=2 {n + k (n− 1)}|an| < 1. Putting α = 0,A = 1 − 2β, B = −1 with 0 ≤ β < 1 in Theorem 1, we have the result below which is comparable to the one obtained by Shams et al. [22]. Corollary 3. A function f (z) ∈ SD (k,β) , if it satisfies the condition ∞∑ n=2 {n (k + 1) − (k + β)}|an| < 1 −β. ALPHA CONVEX FUNCTIONS 75 Putting α = 0,A = 1 − 2β, B = −1 with 0 ≤ β < 1 and k = 0 in Theorem 1, we get the following result proved by Silverman [23]. Corollary 4. A function f (z) ∈ S∗ (β) , if it satisfies the condition ∞∑ n=2 {n−β}|an| < 1 −β. Putting α = 1, in Theorem 1, we can obtain Corollary 5, below which is comparable to the result obtained by Noor and Malik [15]. Corollary 5. A function f ∈ k −UCV [A,B] , if it satisfies the condition ∞∑ n=2 n{2 (k + 1) (n− 1) + |n (B + 1) + (A + 1)|} |an| < |B −A| . The following is an inclusion result stating the fact that k −UMα[A,B] ⊂ k −ST[A,B]. Theorem 2. Let f(z) ∈ k −UMα[A,B]. Then f(z) ∈ k −ST[A,B]. Proof. Let f(z) ∈ k −UMα[A,B] and let (3.8) zf ′ (z) f(z) = p(z), where p(z) is analytic in E with p(0) = 1. Differentiating logarithmically we have (3.9) (zf′(z)) ′ f′(z) = p(z) + zp ′ (z) p(z) . Using (3.8) and (3.9), we have J (α,f; z) = p(z) + αzp ′ (z) p(z) . Since f(z) ∈ k −UMα[A,B], so we obtain J (α,f; z) = p(z) + zp ′ (z) 1 α p(z) ∈ k −UMα[A,B]. Since < (( 2k+1−A 2k+1−B ) 1 α ) > 0, z ∈ E, therefore applying Lemma 3, with δ = 1 α and σ = 0, we have zf ′ (z) f(z) = p(z) ≺ pk(A,B,z), which implies that f(z) ∈ k −ST[A,B]. � By giving special values to the parameters in Theorem 2, we get the following well-known result proved by Mocanu in [13]. Corollary 6. Let f(z) ∈ 0 −UMα[1,−1]. Then f(z) ∈ 0 −ST[1,−1]. That is Mα ⊂ S∗, α ≥ 0. 76 NOOR, KHAN AND PIEJKO Theorem 3. If 0 ≤ α1 < α2, then k −UMα2 [A,B] ⊂ k −UMα1 [A,B]. Proof. Let f(z) ∈ k −UMα2 [A,B]. Then consider J (α1,f; z) = [ (1 −α1) zf ′ (z) f(z) + α1 ( 1 + zf ′′ (z) f ′ (z) )] = ( 1 − α1 α2 ) zf ′ (z) f(z) + α1 α2 [ (1 −α2) zf ′ (z) pf(z) + α2 ( 1 + zf ′′ (z) f ′ (z) )] = ( 1 − α1 α2 ) J (0,f; z) + α1 α2 (J (α2,f; z)) . Now as f(z) ∈ k −UMα2 [A,B] so J (α2,f; z) ∈ k −p[A,B], also from Theorem 2, J (0,f; z) ∈ k −p[A,B]. Using theses along with Lemma 2, we have J (α1,f; z) ∈ k −p[A,B], which implies that f(z) ∈ k −UMα1 [A,B]. � Theorem 4. A function f (z) is in k −UMα[A,B],α > 0, if and only if there exists a function g (z) belonging to the class k −ST[A,B], such that (3.10) f (z) =   1 α z∫ 0 {g (z)} 1 α t−1dt  α . Proof. Let us set g (z) = f (z) { zf′(z) f (z) }α , so that (3.10) is satisfied. Logarithmically differentiation gives us zg′(z) g (z) = (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) . Hence f ∈ k −UMα[A,B] if and only if g ∈ k −ST[A,B]. � Theorem 5. Let the function f (z) ∈ k −UMα[A,B]. Then |a2| ≤ (A−B) δk 2 (1 + α) , where δk is given by (1.3). Proof. Let f (z) ∈ k −UMα [A,B] . Then (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) = p (z) z ∈ E, where p (z) ≺ (A + 1) pk (z) − (A− 1) (B + 1) pk (z) − (B − 1) = 1 + 1 2 (A−B) δkz + · · · , where pk (z) = 1 + δkz + · · · . ALPHA CONVEX FUNCTIONS 77 Now using the definition of subordination we can see that there exists a function ω (z) analytic in E with ω (0) = 0 and |ω (z)| < 1 such that (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) = 1 + 1 2 (A−B) δkω (z) + · · · 1 + (1 + α) a2z + ( 2 (1 + 2α) a3 − (1 + 3α) a22 ) z2 · · · = 1 + 1 2 (A−B) δk ( c1z + c2z 2 + · · · ) + · · · . Comparing the coefficient of z both sides and using well known result due to Janowski and Lemma 1, we have |(1 + α) a2| ≤ 1 2 (A−B) δk. This gives |a2| ≤ (A−B) δk 2 (1 + α) and the proof is complete. � Taking α = 0,A = 1, B = −1 in Theorem 5, we can obtain the following result proved in [5]. Corollary 7. Let f ∈ k −ST . Then |a2| ≤ δk, where δk is given by (1.3). Putting k = 0,δk = 2, α = 0,A = 1,B = −1 in Theorem 5, we can obtain Corollary 8 below which is the result obtained in [2]. Corollary 8. Let f ∈ S∗. Then |a2| ≤ 2. Putting α = 1,A = 1, B = −1 in Theorem 5, we can obtain Corollary 9 below which is comparable to the result obtained in [4]. Corollary 9. Let f ∈ k −UCV . Then |a2| ≤ δk 2 , where δk is given by (1.3). Putting k = 0,δk = 2, α = 0,A = 1, B = −1 in Theorem 5, we can obtain Corollary 10 below which is the result obtained in [2]. Corollary 10. Let f ∈C. Then |a2| ≤ 1. Theorem 6. The range of every univalent functions f ∈ k −UMα[A,B], contains the unit disk Rα,δk = 2 (1 + α) 4 (1 + α) + (A−B) δk , where δk is given by (1.3). Proof. Let ω◦ be any complex number such that f (z) 6= ω◦. Then ω◦f (z) ω◦ −f (z) = z + ( a2 + 1 ω◦ ) z2 + · · · , is univalent in E so that ∣∣∣∣a2 + 1ω◦ ∣∣∣∣ ≤ 2. Therefore ∣∣∣∣ 1ω◦ ∣∣∣∣ ≤ 4 (1 + α) + (A−B) δk2 (1 + α) . 78 NOOR, KHAN AND PIEJKO Hence using Theorem 5, we have |ω◦| ≤ 2 (1 + α) 4 (1 + α) + (A−B) δk = Rα,δk. � Putting α = 0,A = 1,B = −1 in Theorem 6, we can obtain Corollary 11. Corollary 11. The range of every univalent functions f ∈ k −ST contains the unit disk Rδk = 1 2 + δk , where δk is given by (1.3). Putting α = 1,A = 1,B = −1 in Theorem 6, we can obtain Corollary 12. Corollary 12. The range of every univalent functions f ∈ k −UCV contains the unit disk Rδk = 2 4 + δk , where δk is given by (1.3). Letting k = 1,A = 1 and B = −1, we have the following Theorem. Theorem 7. Let f ∈ UMα and let it be of the form f (z) = z + ∞∑ n=m+1 anz n am+1 6= 0. Then f (z) is strongly starlike of order θ◦, where (3.11) θ◦ = min θ∈(0,1) { 1 − 2xθ cos ( θπ 2 ) + ( αm ( x2 + 1 ) θ 2x + xθ sin ( θπ 2 )) ≥ 0 for all x > 0 } . Proof. From the assumption we have (3.12) < { (1 −α) zf′(z) f (z) + α (zf′(z)) ′ f′(z) } > ∣∣∣∣(1 −α) zf′(z)f (z) + α(zf ′(z)) ′ f′(z) − 1 ∣∣∣∣ . Let p (z) = zf′(z) f(z) , then by p (z) has of the form p(z) = 1 + ∞∑ n=m cnz n, and (3.12) , becomes (3.13) < { p(z) + α z◦p ′(z◦) p(z) } > ∣∣∣∣p(z) + αz◦p′(z◦)p(z) − 1 ∣∣∣∣ . If there exists a point z◦, |z◦| < 1, such that |arg{p(z)}| < π 2 θ for |z| < |z◦| , and |arg p(z◦)| = π 2 θ. Then, applying Lemma 4, we have z◦p ′(z◦) p(z◦) = ilθ, where (p(z◦)) 1 θ = ±ix (x > 0), l ≥ m 2 (x + 1 x ) when arg{p(z◦)} = π 2 θ, and l ≤− m 2 (x + 1 x ) when arg{p(z◦)} = − π 2 θ. ALPHA CONVEX FUNCTIONS 79 Therefore, for the case arg{p(z◦)} = π2 θ, we have (3.14) < ( p(z◦) + α z◦p ′(z◦) p(z◦) ) = < { (ix) θ + αlθ } = xθ cos ( θπ 2 ) , and ∣∣∣∣p(z◦) + αz◦p′(z◦)p(z◦) − 1 ∣∣∣∣ = ∣∣∣(ix)θ + iαlθ − 1∣∣∣ = ∣∣∣∣xθ cos ( θπ 2 ) − 1 + i ( αlθ + xθ sin ( θπ 2 ))∣∣∣∣ = √( xθ cos ( θπ 2 ) − 1 )2 + ( αlθ + xθ sin ( θπ 2 ))2 . (3.15) From (3.11) and then from l ≥ m 2 (x + 1 x ) for θ ≥ θ◦, we have 0 ≤ 1 − 2xθ cos ( θπ 2 ) + ( αθm 2x (x2 + 1) + xθ sin ( θπ 2 ))2 ≤ 1 − 2xθ cos ( θπ 2 ) + ( αθl + xθ sin ( θπ 2 ))2 .(3.16) Therefore, (3.17) 0 ≤ 1 − 2xθ◦ cos ( θ◦π 2 ) + ( αθ◦l + x θ◦ sin ( θ◦π 2 ))2 , by (3.14) and (3.15) is equivalent to the inequality < { p(z◦) + α z◦p ′(z◦) p(z◦) } ≤ ∣∣∣∣p(z◦) + αz◦p′(z◦)p(z◦) − 1 ∣∣∣∣ , which contradicts with (3.11). Therefore, |arg{p(z◦)}| < π2 θ◦ for |z| < 1. For the case arg{p(z◦)} = −π2 θ◦, applying the same method as the above we will get a contradiction. In this way we have proved that f is strongly starlike of order θ◦. This completes the proof. � Letting α = 1, in Theorem 7, we have the result 13 below which is comparable to the one obtained in [18]. Corollary 13. Let f ∈ UCV and let it be of the form f (z) = z + ∞∑ n=m+1 anz n am+1 6= 0. Then f (z) is strongly starlike of order θ◦, where θ◦ = min θ∈(0,1) { 1 − 2xθ cos ( θπ 2 ) + ( m ( x2 + 1 ) θ 2x + xθ sin ( θπ 2 )) ≥ 0 for all x > 0 } . Acknowledgement The authors express deep gratitude to Dr. S. M. Junaid Zaidi, Rector, CIIT, for his support and providing excellent research facilities. This research is carried out under the HEC project grant No. NRPU No. 20-1966/R&D/11-2553. References [1] D. A. Brannan., W. E. Kirwan., On some classes of bounded univalent functions, J. London. Math. Soc. 1(2)(1969), 431-443. [2] P. L. Duren., Univalent functions Grundlehren der Math. Wissenchanften, Springer-Verlag, New York-Berlin (1983). [3] W. Janowski., Some external problem for certain families of analytic functions, I. Ann. Polon. Math. 28(1973),298- 326. [4] S. Kanas, A. Wísniowska., Conic regions and k-uniform convexity, J. Comput. Appl. Math. 105(1999), 327-336. 80 NOOR, KHAN AND PIEJKO [5] S. Kanas, A. Wísniowska., Conic domains and starlike functions, Rev. Roumaine Math. Pure. Appl. 45(2000), 647-657. [6] S. Kanas., Coefficient estimates in subclasses of the Caratheodory class related to conical domains, Acta. Math. Univ. Comenian. 74(2)(2005), 149-161. [7] S. Kanas., Alternative characterization of the class k − UCV and related classes of univalent functions, Serdica Math. J. 25(1999), 341-350. [8] S. Kanas., Techniques of the differential subordination for domains bounded by conic sections, Int. J. Math. Math. Sci. 38(2003), 2389-2400. [9] E. Lindelöf., Mémorie sur certaines inequalitiés dans la théorie des fonctions monogènes et sur quelques propiétés nouvelles de ces fonctions dans le voisinage d’un point singulier essentiel, Acta Soc. Sci. Fenn. 35(1908) 1-35. [10] J. E. Littelwood., On inequalities in the theory of functions, Proc. London. Math. Soc. 23(1925), 481-519. [11] M. S. Liu., On certain subclass of analytic functions; J. South. China Normal Univ (in chinese). 4(2002), 15-20. [12] S. S. Miller., P. T. Mocanu., Differential subordination and univalent functions, Michigan. Math. J. 28(2)(1981), 157-172. [13] P. T. Mocanu., Une propriete de convexite generlise dans la theorie de la representation conforme, Mathematica (Cluj). 11(1969), 127-133. [14] S. Nawaz., Certain subclasses of analytic functions associated with conic domains. Ph.D Thesis(2012), Comsats Institute of Information Technology Islamabad, Pakistan. [15] K. I. Noor, S. N. Malik., On coefficient inequalities of functions associated with conic domains, Comput. Math. Appl. 62(2011), 2209-2217. [16] K. I. Noor, S. N. Malik., M. Arif, M. Raza., On bounded boundary and bounded radius rotation related with Janowski function, World. Appl. Sci. J. 12(6) (2011), 895-902. [17] K. I. Noor, M. Arif, W. Ul-Haq., On k-uniformly close-to-convex functions of complex order, Appl. Math. Comput. 215(2)(2009), 629-635. [18] M. Nunokawa, J. Sokol., On order of strongly starlikeness in the class of uniformly convex functions, Math. Nachr. 288(2015), 1003C1008. [19] W. Rogosinski., On the coefficients of subordinate functions, Proc. Lond. Math. Soc. 48(2)(1943), 48-82. [20] W. Rogosinski., On subordinate functions, Proc. Camb. Phil. Soc. 35(1939), 1-26. [21] M. A. Rosihan., V. Ravichandran., Convolution and Differential subordination for mulitivalent functions, Bull. Malays. Math. Sci. 32(3) (2009), 351-360. [22] S. Shams, S. R. Kulkarni, J. M. Jahangiri., Classes of uniformly starlike and convex functions, Int. J. Math. Math. Sci. 55(2004), 2959-2961. [23] H. Silverman., Univalent functions with negative coefficients, Proc. Amer. Math. Soc. 51(1975), 109-116. [24] J. Stankiewicz., Quelques probl‘emes extr‘emaux dans les classes des fonctions α−angulairement ‘etoil‘ees, Ann. Univ. Mariae Curie–Sk lodowska Sect. A 20(1966), 59-75. 1Department of Mathematics COMSATS Institute of Information Technology, Park Road, Islamabad, Pakistan 2Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, 35-959 Rzeszów, Poland ∗Corresponding author: piejko@prz.edu.pl