Janowski type close-to-convex functions associated with conic regions

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1 Mahmood et al. Journal of Inequalities and Applications ( :259 DOI /s R E S E A R C H Open Access Janowski type close-to-convex functions associated with conic regions Shahid Mahmood 1*, Muhammad Arif 2 and Sarfraz Nawaz Malik 3 * Correspondence: shahidmahmood757@gmail.com 1 Department of Mechanical Engineering, Sarhad University of Science & I. T, Landi Akhun Ahmad, Hayatabad Link. Ring Road, Peshawar, Pakistan Full list of author information is available at the end of the article Abstract The analytic functions, mapping the open unit disk onto petal and oval type regions, introduced by Noor and Malik (Comput. Math. Appl. 62: , 2011, are considered to define and study their associated close-to-convex functions. This work includes certain geometric properties like sufficiency criteria, coefficient estimates, arc length, the growth rate of coefficients of Taylor series, integral preserving properties of these functions. MSC: 30C45; 30C50 Keywords: close-to-convex functions; Janowski functions; conic domain 1 Introduction and definitions Let A be the class of functions f of the form f (z=z + a n z n, (1.1 which are analytic in the open unit disk E = {z C : z <1}.Furthermore,S represents the class of all functions in A which are univalent in E. The convolution (Hadamard product of functions f, g A is defined by (f g(z=z + a n b n z n, z E, where f (zisgivenby(1.1and g(z=z + b n z n, z E. For two functions f and g analytic in E,wesaythatf is subordinate to g,denotedbyf g, if there exists a Schwarz function w with w(0 = 0 and w(z <1suchthatf (z=g(w(z. In particular, if g is univalent in E, thenf (0 = g(0 and f (E g(e. For more details, see [2]. The Author(s This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 2 of 14 Afunctionp analytic in E and of the form p(z=1+ p n z n n=1 belongs to the class P[A, B]ifandonlyif p(z 1+Az, 1 B < A 1. 1+Bz This class was introduced and investigated by Janowski [3]. In particular, if A =1andB = 1, we obtain the class P of functions with a positive real part (see [4, 5]. The classes P and P[A, B] are connected by the relation p(z P (A +1p(z (A 1 P[A, B]. (B +1p(z (B 1 Now consider, for k 0, the classes k CV and k ST of k-uniformly convex functions and corresponding k-starlike functions, respectively, introduced by Kanas and Wisniowska, respectively. For some details, see [6 8]. Consider the domain k = { u + iv; u > k (u v 2}. (1.2 For fixed k, k represents the conic region bounded successively by the imaginary axis (k = 0, the right branch of a hyperbola (0 < k < 1, a parabola (k = 1 and an ellipse (k >1. This domain was studied by Kanas [6 8]. The function p k,withp k (0 = 1, p k (0 > 0 plays theroleofextremalandisgivenby 1+z 1 z, k =0, 1+ 2 (log 1+ z π p k (z= 2 1 z 2, k =1, 1+ 2 sinh 2 [( 2 1 k 2 π arccos k arctan h z], 0 < k <1, 1+ 1 sin[ u(z π t 1 k 2 1 2R(t 0 dx]+ 1, k >1, 1 x 2 1 (tx 2 k 2 1 (1.3 where u(z = z t 1, t (0, 1, z E and t is chosen such that k = cos h( πr (t, with R(t tz 4R(t is Legendre s complete elliptic integral of the first kind and R (t isthecomplementary integral of R(t (see[7 9]. Let P pk denote the class of all those functions p(z whichare analytic in E with p(0 = 1 and p(z p k (z forz E. Clearly, it can be seen that P pk P, where P is the class of functions with a positive real part (see [4, 5]. For the applications and exclusive study of the class P, we refer to [10 14]. More precisely ( k P pk P P, 1+k and, for p P pk,wehave arg p(z λπ 2,

3 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 3 of 14 where λ = 2 π arctan ( 1 k. (1.4 Therefore, we can write p(z=h λ (z, h P. Definition 1.1 ([1] A function p analytic in E belongs to the class k P[A, B]ifandonly if p(z (A +1p k(z (A 1 (B +1p k (z (B 1, k 0, where p k (z isdefinedby(1.3 and 1 B < A 1. Geometrically, the function p(z k P[A, B] takes all values in the domain k [A, B], 1 B < A 1, k 0, which is defined as follows: { ( } (B 1w(z (A 1 k [A, B]= w : > k (B 1w(z (A 1 (B +1w(z (A +1 (B +1w(z (A +1 1 or equivalently k [A, B]= { u + iv : [( B 2 1 ( u 2 + v 2 2(AB 1u + ( A 2 1 ] 2 > k 2[( 2(B +1 ( u 2 + v 2 +2(A + B +2u 2(A (A B 2 v 2]}. The domain k [A, B] retains the conic domain k inside the circular region defined by 0 [A, B]= [A, B]. The impact of [A, B] on the conic domain k changes the original shape of the conic regions. The ends of hyperbola and parabola get closer to each other but never meet anywhere and the ellipse gets the shape of oval. When A 1, B 1, the radius of the circular disk defined by [A, B] tends to infinity; consequently, the arms of hyperbola and parabola expand and the oval turns into ellipse. Definition 1.2 ([1] A function f A is said to be in the class k CV[C, D], 1 D < C 1, if it satisfies the condition ( (D 1 (zf (z f (C 1 (z (D +1 (zf (z f (C +1 (z (D 1 (zf (z > k f (C 1 (z (D +1 (zf (z f (C +1 1 (z (k 0; z E, equivalently, we can write (zf (z k P[C, D]. f (z

4 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 4 of 14 Definition 1.3 ([1] The class k ST [C, D], 1 D < C 1, is the family of all those functions f A such that ( (D 1 zf (z (C 1 f (z (D +1 zf (z (C +1 f (z (D 1 zf (z (C 1 > k f (z (D +1 zf (z (C +1 1 f (z (k 0; z E, or equivalently zf (z f (z k P[C, D]. These two classes were recently introduced by Noor and Malik [1]. Motivated by the recent work presented by Noor and Malik [1],wedefinesomeclasses of analytic functions associated with conic domains as follows. Definition 1.4 Let f A. Thenf k UK[A, B, C, D] if and only if there exists g k ST [C, D]suchthat ( (B 1 zf (z (A 1 g(z (B +1 zf (z (A +1 g(z or equivalently zf (z g(z k P[A, B], where 1 D C 1and 1 B < A 1. (B 1 zf (z (A 1 > k g(z (B +1 zf (z (A +1 1 (k 0, g(z Definition 1.5 Let f A. Thenf k UQ[A, B, C, D] if and only if, for 1 D < C 1, 1 B < A 1andk 0, there exists g k CV[C, D]suchthat ( (B 1 (zf (z g (A 1 (z (B +1 (zf (z g (A +1 (z or equivalently (zf (z k P[A, B]. g (z It can easily be seen that (B 1 (zf (z > k g (A 1 (z (B +1 (zf (z g (A +1 1, (z f k UQ[A, B, C, D] zf k UK[A, B, C, D]. (1.5 Special cases: i. 0 UK[A, B, C, D]=K[A, B, C, D] and 0 UQ[A, B,1, 1]=Q[A, B],subclassesof close-to-convex and quasi-convex functions studied by Silvia and Noor, respectively, see [15, 16].

5 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 5 of 14 ii. k UK[1, 1, 1, 1] = k UK and k UQ[1, 1, 1, 1] = k UQ,theclassof k-uniformly close-to-convex and the class of k-uniformly quasi-convex functions studied by Acu [17]. iii. k UK[1 2β, 1, 1 2γ, 1] = k UK(β, γ and k UQ[1 2β, 1, 1 2γ, 1] = k UQ(β, γ, the well-known class of k-uniformly close-to-convex and the class of k-uniformly quasi-convex functions of order β and type γ,see[18]. iv. 0 UK[1 2β, 1, 1 2γ, 1] = K(β, γ and 0 UQ[1 2β, 1, 1 2γ, 1] = Q(β, γ, well-known classes of close-to-convex and quasi-convex functions of order β type γ,see[19, 20]. v. 0 UK[1, 1, 1, 1] = K and 0 UQ[1, 1, 1, 1] = Q, the classes of close-to-convex and quasi-convex functions; for details, see [21, 22]. Throughout this paper, we assume that 1 D < C 1, 1 B < A 1andk 0unless otherwise specified. 2 A set of lemmas To prove our main results, we need the following lemmas. Lemma 2.1 ([23] Let p(z=1+ n=1 p nz n F(z=1+ n=1 d nz n in E. If F(z is univalent in E and F(E is convex, then p n d 1, n 1. Lemma 2.2 ([1] Let p(z=1+ n=1 c nz n k P[A, B]. Then c n δ(a, B, k, where and δ(a, B, k= (A Bδ k, ( (cos 1 k 2, 0 k <1, π 2 (1 k 2 δ k = 8, k =1, π 2 π 2 4, k >1. t(k 2 1R 2 (t(1+t (2.2 Lemma 2.3 ([2] Let f and g be in the class C and S, respectively. Then, for every function F(z analytic in E with F(0 = 1, we have f (z g(zf(z f (z g(z co ( F(E, z E, where denotes the well-known convolution of two analytic functions and cof(e denotes the closed convex hull F(E.

6 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 6 of 14 Lemma 2.4 ([1] Let g k ST [C, D] with k 0 and be given by Then g(z=z + n 2 b n j=0 b n z n. (C Dδ k 2jD, 2(j +1 where δ k is defined by ( The main results and their consequences This section is about the main results of our defined families k UK[A, B, C, D] andk UQ[A, B, C, D]. These families will be thoroughly investigated by studying their important properties including coefficient inequalities, sufficient condition, necessary condition, arc length problem, the growth rate of coefficients, convolution preserving properties and the radius of convexity problem. We will also discuss some special cases of our main results. 1. Coefficient inequalities Theorem 3.1 Let f k UK[A, B, C, D], and let it be of the form given by (1.1. Then, for n 2, we have a n 1 n n 2 (C Dδ k 2iD 2(i +1 + (A B δ k 2n where δ k is defined by (2.2. This result is not sharp. Proof Let us take n 1 j 2 j=1 (C Dδ k 2iD, 2(i +1 zf (z=g(zp(z, (3.1 where p k P[A, B]andg k ST [C, D]. Let zf (z=z+ na nz n, g(z=z+ b nz n and p(z=1+ n=1 c nz n.then(3.1becomes ( z + na n z n = z + b n z (1+ n c n z. n Equating the coefficients of z n on both sides, we have n 1 na n = b n + b j c n j. This implies that j=1 n=1 n 1 n a n b n + b j c n j. (3.2 j=1

7 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 7 of 14 Since p k P[A, B] andg k ST [C, D], therefore by Lemma 2.2 and Lemma 2.4 we have and n 2 b n (C Dδ k 2iD 2(i +1 c n 1 2 (A B δ k. Hence (3.2becomes n 2 n a n which implies that (C Dδ k 2iD 2(i +1 n 1 + j=1 (A B δ k 2 j 2 (C Dδ k 2iD, 2(i +1 a n 1 n n 2 (C Dδ k 2iD 2(i +1 + (A B δ k 2n n 1 j 2 j=1 (C Dδ k 2iD. 2(i +1 This completes the proof. Corollary 3.2 ([9] Let f k UK[1, 1, 1, 1] which has the form (1.1. Then, for n 2, a n ( δ k n 1 n! + δ k n n 1 j=0 ( δ k j 1 (j 1!. Corollary 3.3 Let f k UK[1 2β, 1, 1 2γ, 1] = f k UK[β, γ ] which has the form (1.1. Then, for n 2, a n ( δ k n 1 n! + δ k n n 1 j=0 ( δ k j 1 (j 1!. Corollary 3.4 ([21] Let f 0 UK[1, 1, 1, 1] = K which has the form (1.1. Then, for n 2, a n n. Using relation (1.5and Theorem 3.1, we obtain immediatelythe followingresult. Theorem 3.5 Let f k UQ[A, B, C, D] which has the form (1.1. Then, for n 2, a n 1 n 2 n 2 (C Dδ k 2iD 2(i +1 + (A B δ k 2n 2 n 1 j 2 j=1 (C Dδ k 2iD. 2(i +1

8 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 8 of 14 By assigning different permissible values to the parameters, we obtain several known results, see [17, 21, 22]. 2. Sufficient conditions Theorem 3.6 Let f A and be given by (1.1. Then f k UK[A, B, C, D] if [ 2(k +1 bn na n + ] (B +1na n (A +1b n < B A. (3.3 Proof Let us assume that equation (3.3 holds true. It is sufficient to show that (B 1 zf (z (A 1 ( k g(z (B +1 zf (z (A +1 1 (B 1 zf (z (A 1 g(z (B +1 zf (z (A +1 1 <1. g(z g(z Now consider Since (B 1 zf (z (A 1 g(z (B +1 zf (z (A +1 1 = (B 1zf (z (A 1g(z (B +1zf (z (A +1g(z 1 g(z =2 g(z zf (z (B +1zf (z (A +1g(z =2 (b n na n z n (B Az + [(B +1na n (A +1b n ]z n 2 b n na n (B A (B +1na n (A +1b n. (B 1 zf (z (A 1 ( k g(z (B +1 zf (z (A +1 1 (B 1 zf (z (A 1 g(z (B +1 zf (z (A +1 1 g(z g(z (B 1 zf (z (A 1 (k +1 g(z (B +1 zf (z (A (k +1 b n na n (B A g(z (B +1na n (A +1b n. The last inequality is bounded by 1 if 2(k +1 b n na n B A (B +1nan (A +1b n. Hence we have [ 2(k +1 bn na n + ] (B +1na n (A +1b n B A. This completes the proof.

9 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 9 of 14 Theorem 3.7 Let f A which has the form (1.1. Then f k UQ[A, B, C, D] if n [ 2(k +1 b n na n + ] (B +1na n (A +1b n < B A. The prooffollowsimmediately by using Theorem 3.6 and relation (1.5. Corollary 3.8 ([24] A function is said to be in the class 1 UQ[1 2β, 1, 1, 1] = UQ(β for g(z=zif n 2 a n < 1 β Necessary condition Theorem 3.9 Let f k UK[A, B, C, D]. Then, for < θ 2, z E, { (zf (z } [ ] C D dθ > f (z 2k +1 D + λ π, where λ is defined by (1.4. Proof Since f k UK[A, B, C, D], there exists g k CV[C, D] C(β 1, β 1 = 2k +1 C 2k +1 D (3.4 such that f (z=g (zp(z, where p k P[A, B]. We can write f (z= ( g 1 (z 1 β 1 h λ (z, h P[A, B] P, g 1 C. For z = re iθ,0 r <1,0 θ 2 2π,wehave { (zf (z } dθ =(1 β f 1 (z + λ { (zg 1 (z g 1 (z { zh (z h(z } dθ + β 1 (θ 2 } dθ. (3.5 Also,weobservethat,forh P[A, B], θ arg h( re iθ = θ { i ln h ( re iθ} { re iθ h (re iθ } =. h(re iθ

10 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 10 of 14 Therefore { re iθ h (re iθ } dθ = arg h ( re iθ 2 arg h ( re iθ 1, h(re iθ this implies that max h P[A,B] Since h P[A, B], so h(z 1 ABr2 1 B 2 r 2 From (3.6, we observe that max h P[A,B] Also, for g 1 C,wehave { re iθ h (re iθ } dθ h(re iθ = max ( ( arg h re iθ 2 arg h re i. (3.6 h P[A,B] (A Br 1 B 2 r 2. { re iθ h (re iθ } ( dθ (A Br h(re iθ 2 sin 1 1 ABr 2 { (zg 1 (z } g 1 (z dθ > π. π 2cos 1 ( (A Br 1 ABr 2. (3.7 Using (3.6 and(3.7 in(3.5, we obtain { (zf (z f (z } ( (A Br dθ > (1 β 1 π + β 1 (θ 2 λπ +2λcos 1 1 ABr 2 [ ] (C D > 2k +1 D + λ π(r 1. This completes the required result. mark 3.10 Let f k UK[A, B, C, D]. Then, for { (zf (z } dθ > π, f (z C D 2k+1 D <1 λ, and hence f is univalent in E,see[21]. Corollary 3.11 ([21] Let f 0 UK[1, 1, 1, 1]. Then, for < θ 2, z E, { (zf (z } dθ > π. f (z

11 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 11 of Arc length problem Theorem 3.12 Let f k UK[A, B, C, D] which has the form (1.1. Then ( 1 2(1 β1 1 L r (f C(λ, C, D 1 r (r 1, where β 1 is defined by (3.4 and C(λ, A, B is a constant depending upon λ, AandB. Proof Let zf (z=g(zh λ (z, (3.8 where g k ST [C, D]andh P[A, B] P.Sincek ST [C, D] S (β 1, see [1]. We can write ( g1 (z 1 β1 g(z=z, g 1 S. z Equation (3.8gives zf (z=z β 1 ( g 1 (z 1 β 1 h λ (z. Now, for z = re iθ, L r (f = 2π 0 zf (z dθ = 2π Using Holder s inequality, we have ( 1 L r (f 2π 2π 2π Since h P[A, B] P,so 1 2π 2π z β ( 1 g 1 (z 1 β 1 h λ (z dθ. ( g 1 (z (1 β 1 ( 2 2 λ 2 ( 2 λ 1 dθ 2π 2π 0 λ h(z 2 2 dθ. (3.9 h(z 2 dθ 1+{(A B2 1}r 2. ( r 2 Using (3.10 and the distortion result for a starlike function in (3.9, we obtain ( 1 2π L r (f 2π 2π 0 r (1 β 2 1( 2 λ 1 re iθ 4(1 β 1 2 λ ( 1 2(1 β1 +λ 1 C(λ, A, B 1 r 2 λ 2 ( 1+{(A B 2 1}r 2 dθ 1 r 2 (r 1, λ 2 where C(λ, A, B=π λ 2 (A B λ and 2(1 β 1 +λ > 1. This completes the proof.

12 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 12 of 14 Corollary 3.13 ([25] Let f 0 UK[1, 1, 1, 1]. Then, for 0<r <1, ( L r (f O M(r log 1 1 r as r 1, where O-notation denotes that the constant is absolute and M(r=max z =r f (z. 5. Growth rate of coefficients Theorem 3.14 Let f k UK[A, B, C, D] which has the form (1.1. Then, for k 0, we have a n C(λ, A, Bn 2(1 β 1+λ 2, where β 1 is defined by (3.4. Proof From Cauchy s theorem with z = re iθ,onecaneasilyhave na n = 1 2π zf (ze inθ dθ. 2πr n 0 This implies that na n 1 2πr n = 2π 0 1 2πr n L r(f. zf (z dθ Using Theorem 3.12 and putting r =1 1, we obtain the required result. n 6. Convolution properties Theorem 3.15 If f k ST [C, D] and ϕ C, then ϕ f k ST [C, D]. Proof To prove the result, we need to prove Consider z(ϕ(z f (z ϕ(z f (z k P[A, B]. z(ϕ(z f (z ϕ(z f (z = = ϕ(z zf (z f (z f (z ϕ(z f (z ϕ(z (zf (z, ϕ(z f (z where zf (z f (z = (z k P[C, D]. Applying Lemma 2.3, we obtain the required result. Theorem 3.16 Let f k UK[A, B, C, D] and ϕ C. Then ϕ f k UK[A, B, C, D].

13 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 13 of 14 Proof Since f k UK[A, B, C, D], there exists g k ST [C, D] suchthat zf (z g(z k P[A, B]. It follows from Lemma 2.3 that ϕ g k ST [C, D]. Now z(ϕ(z f (z ϕ(z g(z = ϕ(z zf (z ϕ(z g(z = ϕ(z zf (z g(z g(z ϕ(z g(z = ϕ(z F(zg(z, ϕ(z g(z where F(z k P[A, B]. Applying Lemma 2.3,wehave z E. z(ϕ(z f (z ϕ(z g(z k UK[A, B, C, D]for 7. Radius of convexity problem Theorem 3.17 Let f k UK[A, B, C, D] in E. Then f C for z < r 1, where r 1 = 2 2(1 β 1 +λ(a B+ ([2(1 β 1 +λ(a B] 2 +4[2β 1 1], (3.11 where λ and β 1 are defined by (1.4 and (3.4, respectively. Proof Let zf (z=g(zp(z, where g k ST [C, D]andp k P[A, B]. Since k ST [C, D] S (β 1, it is known [26] that there exists g 1 S such that ( g1 (z 1 β1 g(z=z. z We can write zf (z=z β 1 ( g 1 (z 1 β 1 h λ (z, h P[A, B] P. (3.12 The logarithmic differentiation of (3.12yields (zf (z f (z = β 1 +(1 β 1 zg 1 (z g 1 (z + λzh (z h(z. Using the distortion result for the classes S and P[A, B], we obtain (zf (z f (z β 1 +(1 β 1 1 r λ(a Br 1+r 1 r 2 1+(1 2β 1r 2 [2(1 β 1 +λ(a B]r 1 r 2. (3.13 The right-hand side of (3.13 is positive for z < r 1,wherer 1 is given by (3.11. We note the following cases: (i. For A =1, B = 1, C =1and D = 1, we obtain the radius of convexity problem for the class k UK.

14 Mahmood et al. Journal of Inequalities and Applications ( :259 Page 14 of 14 (ii. For k =0, we have the radius of convexity for the class K[A, B, C, D]. (iii. For A =1, B = 1, C =1, D = 1and k =0, we have the radius of convexity problem for the well-known class of close-to-convex functions studied by Kaplan [21]. Acknowledgements The authors would like to thank the reviewers of this paper for their valuable comments on the earlier version of the paper. They would also like to acknowledge Prof. Dr. Salim ur hman, V.C. Sarhad University of Science & I. T, for providing excellent research and academic environment. Competing interests The authors declare that they have no competing interests. Authors contributions All authors jointly worked on the results and they read and approved the final manuscript. Author details 1 Department of Mechanical Engineering, Sarhad University of Science & I. T, Landi Akhun Ahmad, Hayatabad Link. Ring Road, Peshawar, Pakistan. 2 Department of Mathematics, Abdul Wali Khan University Mardan, Mardan, Pakistan. 3 Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt, Pakistan. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. ceived: 24 March 2017 Accepted: 5 October 2017 ferences 1. Noor, KI, Malik, SN: On coefficient inequalities of functions associated with conic domains. Comput. Math. Appl. 62, ( Miller, SS, Mocanu, PT: Differential Subordinations: Theory and Applications. Pure and Applied Mathematics, vol Dekker, New York ( Janowski, W: Some extremal problems for certain families of analytic functions. Ann. Pol. Math. 28, ( Goodman, AW: Univalent Functions, Vol. I. Polygonal Publishing House, Washington ( Goodman, AW: Univalent Functions, Vol. II. Polygonal Publishing House, Washington ( Kanas, S: Techniques of the differential subordination for domains bounded by conic sections. Int. J. Math. Math. Sci. 38, ( Kanas, S, Wisniowska, A: Conic regions and k-uniform convexity. J. Comput. Appl. Math. 105, ( Kanas, S, Wisniowska, A: Conic domains and starlike functions. v. Roum. Math. Pures Appl. 45, ( Noor, KI, Arif, M, Ul-Haq, W: On k-uniformly close-to-convex functions of complex order. Appl. Math. Comput. 215, ( Noor, KI, Ul-Haq, W, Arif, M, Mustafa, S: On bounded boundary and bounded radius rotations. J. Inequal. Appl. 2009, Article ID ( Noor, KI: On uniformly univalent functions with respect to symmetrical points. J. Inequal. Appl. 2014, 254 ( Raza, M, Malik, SN, Noor, KI: On some inequalities of a certain class of analytic functions. J. Inequal. Appl. 2012, 250 ( Haq, W, Mahmood, S: Certain properties of a class of close-to-convex functions related to conic domains. Abstr. Appl. Anal. 2013, Article ID ( Ul-Haq, W, Noor, KI: A certain class of analytic functions and the growth rate of Hankel determinant. J. Inequal. Appl. 2012, 309 ( Silvia, EM: Subclasses of close-to-convex functions. Int. J. Math. Math. Sci. 3, ( Noor, KI: On some subclasses of close-to-convex functions in univalent functions. In: Srivistava, H, Owa, S (eds. Fractional Calculus and Their Applications. Wiley, London ( Acu, M: On a subclass of n-uniformly close-to-convex functions. Gen. Math. 14, ( Aghalary, R, Azadi, G: The Dziok-Srivastava operator and k-uniformly starlike functions. J. Inequal. Pure Appl. Math. 6(2, Article ID 52 (2005 (electronic 19. Libera, RJ: Some radius of convexity problems. Duke Math. J. 31, ( Noor, KI: On quasi-convex functions and related topics. Int. J. Math. Math. Sci. 2, ( Kaplan, W: Close-to-convex Schlicht functions. Mich. Math. J. 1, ( Noor, KI, Thomas, DK: Quasi convex univalent functions. Int. J. Math. Math. Sci. 3, ( Rogosinski, W: On the coefficients of subordinate functions. Proc. Lond. Math. Soc. 48, ( Subramanian, KG, Sudharasan, TV, Silverman, H: On uniformly close-to-convex function and uniformly quasi-convex function. Int. J. Math. Math. Sci. 48, ( Thomas, DK: On starlike and close-to-convex univalent functions. J. Lond. Math. Soc. 42, ( Noor, KI: On a generalization of uniformly convex and related functions. Comput. Math. Appl. 61(1, (2011