Research Article Differential Subordinations of Arithmetic and Geometric Means of Some Functionals Related to a Sector

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1 International Journal of Mathematics and Mathematical Sciences Volume 2011, Article ID , 19 pages doi: /2011/ Research Article Differential Subordinations of Arithmetic and Geometric Means of Some Functionals Related to a Sector A. Lecko 1 and M. Lecko 2 1 Department of Analysis and Differential Equations, University of Warmia and Mazury in Olsztyn, Street Zolnierska 14, Olsztyn, Poland 2 Department of Mathematics, Rzeszów University of Technology, Street W. Pola 2, Rzeszów, Poland Correspondence should be addressed to A. Lecko, alecko@matman.uwm.edu.pl Received 30 December 2010; Accepted 30 March 2011 Academic Editor: Hans Keiding Copyright q 2011 A. Lecko and M. Lecko. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Some general theorems on differential subordinations of some functionals connected with arithmetic and geometric means related to a sector are proved. These results unify a number of well known results concerning inclusion relation between the classes of analytic functions built with using arithmetic and geometric means. 1. Introduction For r>0letd r {z C : z <r}. Let D D 1. Let the functions f and F be analytic in the unit disc D. Afunctionf is called subordinate to F, written f F, iff is univalent in D, f 0 F 0 and f D F D. Let D be a domain in C 2 and ψ : C 2 D C be an analytic function, and let p be a function analytic in D with p,zp D, z D and h be a function analytic and univalent in D. The function p is said to satisfy the first-order differential subordination if ψ ( p,zp ) h, ψ ( p 0, 0 ) h 0, z D. 1.1 The general theory of the differential subordinations has been studied intensively by many authors. A survey of this theory can by found in the monograph by Miller and Mocanu 1.

2 2 International Journal of Mathematics and Mathematical Sciences For β 0, 2 let h β ( ) 1 z β, z D z It is clear that h β maps univalently D onto the sector of the angle βπ symmetrical with respect to the real axis with the vertex at the origin. In this paper we are interested in the following problem referring to 1.1 to find the constant c k n, γ, α, β so that to the following relation is true: { ] γ } p 1 α zp h ck n,α,γ,β,z p k D p h β, 1.3 with suitable assumptions on function p and constants n, α, γ, β. For selected parameters n, γ, α, β the theorems presented here reduce to the well-known theorems proved by various authors. Particularly, results of this type can be applied to examine inclusion relation between subclasses of analytic functions defined with using arithmetic or geometric means of some functionals, for example, the class of α-convex functions or γ-starlike functions. The lemma below that slightly generalizes a lemma proved by Miller and Mocanu 2 will be required in our investigation. Lemma 1.1 see 2. Let q : D q 0 1.Let C be a function analytic and univalent on D,injectiveon D and p 1 c k z k, z D, 1.4 k n be analytic in D,p/ 1. Suppose that there exists a point z 0 D such that p z 0 q D and p {z C : z < z 0 } q D. 1.5 If ξ 0 q 1 p z 0 and q ξ 0 exists, there exists an m n for which z 0 p z 0 mξ 0 q ξ Main Results In the first theorem which follows directly from Theorem we prove that c k n, α, γ, β β. Let us start with the following definition.

3 International Journal of Mathematics and Mathematical Sciences 3 Definition 2.1 see 3. Letγ 0, 1 and Φ be a function analytic in domain D C.ByH γ, Φ will be denoted the class of functions p analytic in D with p 0 1, p/ 1andp D D such that the function ] Q p 1 zp γ p Φ p, Q 0 1, z D, 2.1 is well defined in D. Theorem 2.2 see 3. Let γ 0, 1,h a convex function such that 0 h D, h 0 1, Φ a function analytic in a domain D C such that h D D, andre Φ h > 0 for z D.If p H γ,φ and p 1 zp p Φ( p ) ] γ h, z D, 2.2 p h. 2.3 Definition 2.3. Let k, n N, α 0andγ 0, 1. By H k n, α, γ will be denoted the class of functions p analytic in D of the form 1.4 such that the function ] γ Q p 1 α zp, Q 0 1, z p k D, 2.4 is well defined in D. Remark Setting we see that Φ w Φ k,α w α, w C \ {0}, 2.5 wk 1 H k ( 1,α,γ ) H ( γ,φk,α ) For each k, n, α, γ as in Definition 2.3 the class H k n, α, γ is nonempty. To see this take p 1 c n z n, z D, 2.7 for sufficiently small c n C. 3 Clearly, for γ 0theclassH k n, α, 0 contains all analytic functions p of the form 1.4.

4 4 International Journal of Mathematics and Mathematical Sciences 4 Let k γ 1. Then Q p αzp, z D. 2.8 Therefore the class H 1 n, α, 1 contains all analytic functions p of the form Let p be analytic function in D of the form 1.4. Suppose that p z 0 0for some z 0 D. Then p z z 0 m p 1, z D, 2.9 where m 1andp 1 is analytic function in D with p 1 z 0 / 0forz D. Then we have z p p k zm z z 0 m 1 p 1 z z 0 m p 1 z z0 m p 1 ] k mp 1 z z 0 p 1 z z 0 m k 1 1 p k Henceweseethatfork 1andγ 0, 1 or for k 2andγ 0, 1 the function ] γ Q p 1 α zp, z p k D, 2.11 has a pole at z 0. Therefore for such k and γ we see that every p H k n, α, γ is nonvanishing in D. Theorem 2.5. Let k N, α 0, γ 0, 1 and β 0, 1 be such that k 1 β 1. If p H k 1,α,γ and ] γ p 1 α zp h p k β, z D, 2.12 p h β Proof. The case α 0 is evident so we assume that α > 0. For k N and α > 0letΦ Φ k,α be defined by 2.5. Forβ 0, 1 the function h β is convex with 0 h D. Since k 1 β 1, we have Re Φ ( h β ) ( ( ) 1 z β ) { ( ) 1 z k 1 β } Re Φ α Re > 0, z D z 1 z Applying Theorem 2.2 with h β instead of h we get the assertion.

5 International Journal of Mathematics and Mathematical Sciences 5 NowweprovetwotheoremswereweimprovetheresultofTheorem 2.5.Theproblem 1.3 will be divided into two cases: k 1andk>1. First we consider the case k 1. The theorem below was proved in 4. Tobeselfcontained we include its proof. Theorem 2.6. Fix n N, α 0 and γ 0, 1. Let β 0,β 1 n, α, γ, where β β 1 n, α, γ is the solution of the equation β c 1 ( n, α, γ, β ) 4 γ, 2.15 with c 1 ( n, α, γ, β ) β 2γ π arctan( nαβ ) If p H 1 n, α, γ and ] p 1 α zp γ h c1 n,α,γ,β, z D, 2.17 p p h β Proof. 1 Assume that α>0andγ 0, 1 since the cases γ 0orα 0 are evident. Suppose, on the contrary, that p is not subordinate to h β. Then, by the minimum principle for harmonic mappings there exists r 0 0, 1 such that p D r0 h β D, 2.19 and one of the following cases hold: or or max { Arg p : z D r0 } max { Arg p : z r0 } β π 2, 2.20 min { Arg p : z D r0 } min { Arg p : z r0 } β π 2, 2.21 p z 0 0, 2.22 for some z 0 D r0.

6 6 International Journal of Mathematics and Mathematical Sciences 2 Assume that 2.20 holds. Then there exists z 0 D r0 such that Arg p z 0 β π Let ξ 0 h 1 β p z 0. Thus ( ) 1 β ξ0 p z 0 h β ξ 0 / ξ 0 Therefore ξ 0 / ± 1and 1 ξ 0 1 ξ 0 xi 2.25 for x>0, that is, ξ 0 xi 1 xi Since ξ 0 / ± 1, so h β ξ 0 exists. Hence and by Lemma 1.1 there exists an m n for which z 0 p z 0 mξ 0 h β ξ Consequently, p z 0 1 α z 0p ] z 0 γ h β ξ 0 1 mα ξ 0h β ξ 0 p z 0 h β ξ 0 ] β xi β 1 mαβ( 1 x 2) γ i]. 2x 2.28 In view of the fact that x>0letustake arg {1 mαβ( 1 x 2) } i 2x 0, π )

7 International Journal of Mathematics and Mathematical Sciences 7 Hence and from 2.28 we have { arg p z 0 1 α z 0p ] z 0 γ } arg { xi 1 β mαβ( 1 x 2) ] γ } i p z 0 2x β π { 2 γ arg 1 mαβ( 1 x 2) } i 2x β π ( ( mαβ 1 x 2 ) ) 2 γ arctan. 2x 2.30 By the above and by the fact that m n we have { arg p z 0 1 α z 0p ] z 0 γ } β π2 ( ( ) ) nαβ 1 x 2 p z 0 γ arctan 2x β π 2 γ arctan( nαβ ) c 1 ( n, α, γ, β )π On the other hand, 2.30 yields { arg p z 0 1 α z 0p ] z 0 γ } β π2 ( ( mαβ 1 x 2 ) ) p z 0 γ arctan i ( β γ )π 2x Finally, the above and 2.31 lead to { ( )π c 1 n, α, γ, β 2 arg p z 0 1 α z 0p ] z 0 γ } p z 0 ( β γ )π 2 2π c ( )π 1 n, α, γ, β 2, 2.33 for all β 0,β 1 n, α, γ. Thus we arrive at a contradiction with 2.17 so p h β. 4 When 2.21 holds, we see that x<0in Nextwefinishtheproofbysimilar argumentations like in the above. 5 Assume now that 2.22 holds. In view of Remark 2.4 this is possible only when k γ 1. a For β<1theboundary h β D has the corner at 0 of the angle βπ < π.sincep D r0 is an analytic curve, in view of 2.19 the case p z 0 0does not hold for β<1. b Let now β 1. Assume that p z 0 / 0. Since z 0 p z 0 is an outer normal to the curve p D r0 at p z 0, by 2.19 we see that ( 3 β ) π π β π 2 arg{ z 0 p z 0 } β π 2 π 2 ( β 1 )π

8 8 International Journal of Mathematics and Mathematical Sciences Hence taking into account that ( ) ( ) 3 β c 1 n, α, 1,β, β 1 4 c1 n, α, 1,β, p z 0 αz 0 p z 0 αz 0 p z 0, 2.35 we deduce that ( )π c 1 n, α, 1,β 2 arg{ p z 0 αz 0 p z 0 } ( β γ )π 2 2π c ( )π 1 n, α, 1,β 2, 2.36 for all β 0,β 1 n, α, γ. In this way we arrive at a contradiction with 2.17 so p h β. If p z 0 0, p z 0 αz 0 p z 0 0 / h β D, 2.37 and once again we contradict Special Cases 1 The case n 1, α 1wasprovedin 5. 2 The case γ 1wasprovedin 6. Corollary 2.7 see 6. Let n N, α 0. Let β 0,β 1 n, α, 1, where β β 1 n, α, 1 is the solution of the equation β c 1 ( n, α, 1,β ) 3, 2.38 with If p is analytic function in D c 1 ( n, α, 1,β ) β 2 π arctan( nαβ ). of the form 1.4 and 2.39 p αzp h c1 n,α,1,β, z D, 2.40 p h β The case γ 1, α 1wasremarked 7. 4 The case γ 1, α 1, n 1 was proved in detail in 8. Now we consider the problem 1.3 for k 2. Theorem 2.8. Let k 2, n N, α 0, γ 0, 1 and let β 0, 1/ k 1. If p H k n, α, γ and ] γ p 1 α zp h ck n,α,γ,β, z p k D, 2.42

9 International Journal of Mathematics and Mathematical Sciences 9 p h β, 2.43 where c k ( n, α, γ, β ) β 2γ π arctan nαβ cos ( k 1 βπ/2 ) ( 1 k 1 β ) 1 k 1 β /2 ( 1 k 1 β ) 1 k 1 β /2 nαβ sin ( k 1 βπ/2 ) ] Proof. 1 We repeat argumentation from Parts 1 and 2 of the proof of Theorem We have p z 0 1 α z ] 0p γ z 0 h p k β ξ 0 1 mα ξ 0h β ξ β 0 z 0 h k β ξ 0 xi β 1 mαβ( 1 x 2) i 2x xi k 1 β xi β 1 mαβ( 1 x 2) ] γ i 1 k 1 β. 2x 1 k 1 β ] γ 2.45 Since x>0and1 k 1 β 0, we can take arg {1 mαβ( 1 x 2) 2x 1 k 1 β i 1 k 1 β } 0, π ) Hence { arg p z 0 1 α z ] 0p γ } z 0 p k z 0 arg { xi β 1 mαβ( 1 x 2) 2x 1 k 1 β i 1 k 1 β ] γ } β π { 2 γ arg 1 mαβ( 1 x 2) ( ) k 1 βπ sin i mαβ( 1 x 2) ( ) } k 1 βπ cos 2x 1 k 1 β 2 2x 1 k 1 β 2 β π ( ( mαβ 1 x 2 ) 2 γ arctan /2x 1 k 1 β) cos ( k 1 βπ/2 ) ] 1 ( mαβ 1 x 2 /2x 1 k 1 β) sin ( k 1 βπ/2 ). 2.47

10 10 International Journal of Mathematics and Mathematical Sciences Thus, from 2.46 and by the fact that m n we obtain { arg p z 0 1 α z ] 0p γ } z 0 β π2 ( ) ] nαβa x cos k 1 βπ/2 p k z 0 γ arctan 1 nαβa x sin ( k 1 βπ/2 ), 2.48 where a x 1 x2 2x, x / k 1 β We have ( ) 1 k 1 β x 2 a ( 1 k 1 β ) x, x/ x 2 k 1 β 3 Assume now that k 1 β <1. Observe that the function a attains its minimum at the point x 0 1 k 1 β 1 k 1 β Moreover a x 0 1 ( 1 k 1 β ) 1 k 1 β /2 ( 1 k 1 β ) 1 k 1 β / Hence, and from 2.48, wehave { arg p z 0 1 α z ] 0p γ } z 0 p k z 0 β π { 2 γ arctan nαβa x0 cos ( k 1 βπ/2 ) } 1 nαβa x 0 sin ( k 1 βπ/2 ) β π 2 nαβ cos ( k 1 βπ/2 ) ] γ arctan ( ) 1 k 1 β /2 ( ) 1 k 1 β /2 ( ) 1 k 1 β 1 k 1 β nαβ sin k 1 βπ/2 c k ( n, α, γ, β )π

11 International Journal of Mathematics and Mathematical Sciences 11 On the other hand, using the fact that 0 k 1 βπ/2 <π/2, from 2.47 we obtain { arg p z 0 1 α z ] 0p γ } z 0 p k z 0 β π ( ( mαβ 1 x 2 ) 2 γ arctan /2x 1 k 1 β) cos ( k 1 βπ/2 ) ] ( mαβ 1 x 2 /2x 1 k 1 β) sin ( k 1 βπ/2 ) ( β γ )π2. Finally, the above and 2.53 yield { ( )π c k n, α, γ, β 2 arg p z 0 1 α z ] 0p γ } z 0 ( β γ )π p k z 0 2 2π c ( )π k n, α, γ, β 2, 2.55 for all β 0, 1/ k 1. Thus we arrive at a contradiction with 2.17 so p h β. 4 For k 1 β 1wehavec k n, α, γ, β β. This ends the proof of the theorem for the case x>0. 5 When 2.21 holds, we see that x<0in Nextwefinishtheproofbysimilar argumentations like in the above. 6 Since β 1/ k 1 < 1, arguing as in Part 5 a of the proof of Theorem 2.6 we see that the case 2.22 does not hold. Special Cases 1 β 1/ k 1. Then c k n, α, γ, β β. Corollary 2.9. Let k 2,n N,α 0 and γ 0, 1. If p H k n, α, γ and ] γ p 1 α zp h p k 1/ k 1, z D, 2.56 p h 1/ k k 2, β 1. Corollary Let n N, α 0 and γ 0, 1. If p H 2 n, α, γ and { ] Re p 1 α zp γ } > 0, z p 2 D, 2.58

12 12 International Journal of Mathematics and Mathematical Sciences Re { p } > 0, z D The case n 1, α 1wasprovedin 9. 4 γ 1. Corollary Let k 2, n N, α 0 and β 0, 1/ k 1. If p is a function analytic in D of the form 1.4 nonvanishing in D and p α zp p k 1 h c k n,α,1,β, z D, 2.60 p h β, 2.61 where c k n, α, 1,β is given by γ 1, k 2. Corollary Let n N, α 0,and β 0, 1. If p is a function analytic in D nonvanishing in D and of the form 1.4 p α zp p h c 2 n,α,1,β, z D, 2.62 where ( ) 2 c 2 n, α, 1,β β π arctan nαβ cos ( βπ/2 ) ( ) 1 β /2 1 ], 1 β β 1 β /2 nαβ sin ( βπ/2 ) 2.63 p h β The case γ 1, k 2, n 1, α 1wasprovedin 10. The same result was reproved in 11 and once again in γ 1, k 2, β 1. Corollary Let n N, α 0, and β 0, 1. If p is an analytic function in D 1.4 nonvanishing in D and of the form { } Re p α zp > 0, z D, 2.65 p

13 International Journal of Mathematics and Mathematical Sciences Applications Re { p } > 0, z D All this type results can be applied in the theory of analytic functions. Some results concerning the inclusion relations between subclasses of analytic functions can be formulated. Let A n, n N, denote the class of functions of the form f z a k z k, k n which is analytic in D. For short, let A A 1. Also let S denote the class of all functions in A which are univalent in D. To use theorems and corollaries listed in the previous section we put instead of the function p some functionals over the class A n, such as p f/z, p zf /f or the others. In this way the inclusion relations between selected subclasses of analytic functions can be obtained Arithmetic Means I γ 1, k 1. i p f/z, z D, f A n, n N. For n N, α 0, β 0, 1 let R n α, β denote class of functions f A n such that 1 α f z αf h β, z D, 3.2 or, equivalently, { arg 1 α f z } αf <β π 2, z D. 3.3 Using Corollary 2.7 we have the following. Corollary 3.1. Let n N, α 0,and β 0,β 1 n, α, 1. If f A n and { arg 1 α f z } αf ( )π <c 1 n, α, 1,β 2, 3.4 { } f arg <β π z 2, z D. 3.5 The above result we can write in the following form.

14 14 International Journal of Mathematics and Mathematical Sciences Corollary 3.2. R n ( α, c1 ( n, α, 1,β )) Rn ( 0,β ). 3.6 ii p f, z D, f A n, n N For n N, α 0, β 0, 1 let T n α, β denote class of functions f A n such that f αzf h β, z D, 3.7 or, equivalently, arg { f αzf } <β π 2, z D. 3.8 Remark The class T 1 1, 1 was introduced in The class T 1 α, 1 coincides with the class H α, 1, 1 studied in 14. Observe that f T n α, β if and only if zf R n α, β. Using Corollary 2.7 we have the following. Corollary 3.4. Let n N, α 0,and β 0,β 1 n, α, 1. If f A n and arg { f αzf } <c1 ( n, α, 1,β )π 2, z D, 3.9 arg { f } <β π 2, z D Hence we have the following. Corollary 3.5. T n ( α, c1 ( n, α, 1,β )) Tn ( 0,β ) II γ 1, k 2. ii p zf /f, z D, f A n, n N. For n N, α 0, β 0, 1 let M n α, β denote class of functions f A n such that ff / 0forz D and 1 α zf ( f α 1 zf ) f h β, z D, 3.12

15 International Journal of Mathematics and Mathematical Sciences 15 or, equivalently, { arg 1 α zf ( f α 1 zf f )} <β π 2, z D Remark The class M 1 α, 1, that is, the class of so-called α-convex functions was introduced by Mocanu The class M 1 0, 1 is identical with the class S of starlike functions. The class M 1 1, 1 is identical with the class C of convex functions. 3 The class M 1 0,β denoted by S β were defined by Brannan and Kirwan 16 and, independently, by Stankiewicz 17, 18. Functions in this class are called strongly starlike of order β. The class M 1 1,β denoted by C β contains functions called strongly convex of order β. Using Corollary 2.12 we have the following result proved by Marjono and Thomas 19. Corollary 3.7. Let n N, α 0,and β 0, 1. If f A n and { arg 1 α zf ( f α 1 zf f )} <c 2 ( n, α, 1,β )π 2, z D, 3.14 { zf } arg <β π f 2, z D For α 1 one has the result due to Nunokawa and Thomas 12]: Corollary 3.8. Let n N and β 0, 1. If f A n and { arg 1 zf } ( )π f <c 2 n, 1, 1,β 2, z D, 3.16 { zf } arg <β, z D f Corollary 3.9. M n ( α, c2 ( n, α, 1,β )) Mn ( 0,β ), M 1 α, 1 S, 3.18 C S.

16 16 International Journal of Mathematics and Mathematical Sciences 3.2. Geometric Mean I α 1, k 1. i p f/z, z D, f A n, n N. For n N, β 0, 1, andγ 0, 1 let L n γ,β denote class of functions f A n such that ( ) f 1 γ ( f ) γ hβ, z D, 3.19 z or equivalently { (f ) 1 γ arg ( f ) } γ z <βπ 2, z D Remark The class L 1 γ,1 was introduced in 20. Applying Theorem 2.6 with α 1 we have the following. Corollary Let n N, γ 0, 1,andβ 0,β 1 n, 1,γ. If f A(1) and { ( ) f 1 γ arg ( f ) } γ z <c ( )π 1 n, 1,γ,β 2, z D, 3.21 { } f arg <β π z 2, z D Corollary L n ( γ,c1 ( n, 1,γ,β )) Ln ( 0,β ) II α 1, k 2. ii p zf /f, z D, f A n, n N. For n N, β 0, 1, andγ 0, 1 let Sn γ,β denote class of functions f A n such that ( zf ) 1 γ ( 1 zf ) γ f f h β, z D, 3.24

17 International Journal of Mathematics and Mathematical Sciences 17 or, equivalently, { ( zf ) arg 1 γ ( f 1 zf f ) γ } <β π 2, z D Remark The class Sn γ,1, that is, the class of so-called introduced by Lewandowski et al Clearly, γ-starlike functions was S 1 0, 1 S, S 1 1, 1 C, S 1( 0,β ) M1 ( 0,β ) S ( β ), 3.26 S 1( 1,β ) M1 ( 1,β ) C ( β ). Using Theorem 2.8 we obtain results due to Darus and Thomas 22. Theorem Let n N, γ 0, 1,and β 0, 1. If f A n and { (zf ) arg 1 γ ( f 1 zf f ) γ } <c 2 ( n, 1,γ,β )π 2, z D, 3.27 { zf } arg <β π f 2, z D Corollary S n( γ; c2 ( n, 1,γ,β )) S n ( β ), S 1( γ,1 ) S As further applications of Theorems 2.6 and 2.8 we can use arbitrary well-defined functionals over the class A n. We recall two examples: 1 p Kδ 1 a f, z D, f A n, n N, a > 0, δ 0, 3.30 z

18 18 International Journal of Mathematics and Mathematical Sciences where the integral operator K δ a over the class A n was defined by Komatu 23 as follows: K δ af where Γ is the Gamma function; 2 aδ Γ δ 1 0 t a 2 ( log 1 t ) δ 1 f zt dt, z D, 3.31 p L λf, z D,f A,λ 1, 3.32 z where the operator L λ over the class A called Ruscheweyh derivative 24 was defined as follows: z L λ f f, z D λ 1 1 z References 1 S. Miller and P. T. Mocanu, Differential Subordinations Theory, Theory and Applications, Marcel Dekker, New York, NY, USA, S. Miller and P. T. Mocanu, Second order differential inequalities in the complex plane, Journal of Mathematical Analysis and Applications, vol. 65, no. 2, pp , A. Lecko, On differential subordinations and Inclusion relation between classes of analytic functions, Complex Variables, vol. 40, pp , A. Lecko, Some differential subordinations depended on real parameters, in Proceeding of the 5th Environment and Mathematics Conference, pp , KUL, Lublin, Poland, A. Lecko, M. Lecko, M. Nunokawa, and S. Owa, Differential subordinations and convex functions related to a sector, Mathematica(Cluj), vol.40 63, no. 2, pp , Y. C. Kim and A. Lecko, Univalence of certain integral operators and differential subordinations, Mathematica(Cluj), vol.43 66, no. 1, pp , P.T. Mocanu, Some simple criteria for starlikeness and convexity, Libertas Math, vol. 13, pp , S. Miller and P. T. Mocanu, Marx-StrohhÄacker differential subordinations systems, Proceedings of the American Mathematical Society, vol. 99, pp , A. Lecko, On differential subordinations connected with convex functions related to a sector, Kyungpook Mathematical Journal, vol. 39, no. 2, pp , P. T. Mocanu, Alpha-convex integral operator and strongly starlike functions, Studia Universitatis Babeş Bolyai, Mathematica, vol. 34, no. 2, pp , M. Nunokawa, On the order of strongly starlikeness of strongly convex functions, Proceedings of the Japan Academy, vol. 69, no. 7, M. Nunokawa and D. K. Thomas, On convex and starlike functions in a sector, Journal of the Australian Mathematical Society. Series A, vol. 60, no. 3, pp , R. Singh and S. Singh, Convolution properties of a class of starlike functions, Proceedings of the American Mathematical Society, vol. 106, pp , Y. Dinggong, Properties of a class of analytic functions, Mathematica Japonica, vol. 41, pp , P. T. Mocanu, Une propriéte de convexité danslareprésentalion conforme, Mathematica (Cluj), vol. 11, no. 34, pp , D. A. Brannan and W. E. Kirwan, On some classes of bounded univalent functions, Journal of the London Mathematical Society, vol. 2, no. 1, pp , J. Stankiewicz, On a family of starlike functions, Annales Universitatis Mariae Curie-Sklodowska. Sectio A, vol , pp ,

19 International Journal of Mathematics and Mathematical Sciences J. Stankiewicz, Quelques problèmes extrémaux dans les classes des fonctionsr -angulairement ètoilées, Annales Universitatis Mariae Curie-Sklodowska. Sectio A, vol. 20, pp , J. Marjono and D. K. Thomas, Subordination on δ-convex functions in a sector, Honam Math Journal, vol. 23, no. 1, pp , S. Kanas, A. Lecko, and J. Stankiewicz, On some classes of the regular functions, Resov, vol. 38, pp , Z. Lewandowski, S. Miller, and E. Zlotkiewicz, Gamma-starlike functions, Annales Universitatis Mariae Curie-Sklodowska, Sectio A, vol. 28, pp , M. Darus and D. K. Thomas, α-logarithmically convex functions, Indian Journal of Pure and Applied Mathematics, vol. 29, no. 10, pp , Y. Komatu, On analytic prolongation of a family of operators, Mathematica(Cluj), vol. 32, no. 55, pp , S. Rusheweyh, New criteria for univalent functions, Proceedings of the American Mathematical Society, vol. 49, pp , 1975.

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