ON STRONG ALPHA-LOGARITHMICALLY CONVEX FUNCTIONS. 1. Introduction and Preliminaries

Transcription

1 ON STRONG ALPHA-LOGARITHMICALLY CONVEX FUNCTIONS NIKOLA TUNESKI AND MASLINA DARUS Abstract. We give sharp sufficient conditions that embed the class of strong alpha-logarithmically convex functions into the class of strong starlike functions. 1. Introduction and Preliminaries Let A denote the class of analytic functions f in the unit disc U = {z : z < 1} and normalized so that f0) = f 0) 1 = 0. A function f A is said to be strongly starlike of order γ, i.e., to belong to S γ), 0 < γ 1, if and only if arg zf < γπ f 2, for all z U. Then S = S 1) is the class of starlike functions in the unit disk U having representation Re zf f functions which consists of functions f A such that > 0, z U. Further, K is the class of convex { Re 1 + zf } f > 0, for all z U. Both of these classes are subclasses of univalent functions in U and moreover K S [1]). Here, we will study the differential operator zf ) α Iα, f; z) 1 + zf f f ) 1 α and give sufficient conditions involving it that imply strong starlikeness of order γ of the function f, i.e., f S γ). Namely, for the class of strong α-logarithmically 2000 Mathematics Subject Classification. 30C45. Key words and phrases. starlike of order α, sharp criteria, Jack lemma. 1

2 2 NIKOLA TUNESKI AND MASLINA DARUS convex functions of order β, 0 α 1, 0 < β 1, defined by M α β) = {f A : arg Iα, f; z) < βπ/2, z U}, we will find and for the class β α, γ) = sup{β : M α β) S γ)}, M α λ) = {f A : Iα, f; z) 1 < λ, z U}, we will obtain λ α, γ) = sup{λ : M α λ) S γ)}. In that purpose we will need the well known Jack Lemma. Lemma 1. [2] Let ω be a non-constant and analytic function in the unit disk U with ω0) = 0. If ω attains its maximum value on the circle z = r at the point z 0 then z 0 ω z 0 ) = kωz 0 ) and k Main Results and Consequences Using Jack Lemma we receive the following useful result. Lemma 2. Let p be an analytic function in the unit disk U such that p0) = 1, and let 0 α 1 and 0 < γ 1. If 1) zp [p ) α ] arg p < α + γ αγ) π 2, z U, then arg p < γπ/2 for all z U. This result is sharp. Proof. Let define a function ω by p = U, ω0) = 0 and zp ) α p p = ) [ γ 1 + ω 2γ 1 ω 1+ω 1 ω) γ. Then ω is analytic in ) ] 1 γ α 1 + ω zω 1 ω 1 ω) It is easy to realize that the conclusion of the lemma, arg p < γπ/2 for all z U, is equivalent to ω < 1 for all z U. Therefore, assume the contrary: there exists

3 ON STRONG ALPHA-LOGARITHMICALLY CONVEX FUNCTIONS 3 a z 0 U such that ωz 0 ) = 1. Then by the Jack lemma z 0 ω z 0 ) = kωz 0 ) and k 1. For such z 0, taking 1+ωz 0) 1 ωz 0) = ix, we have z0 p ) α z 0 ) pz 0 ) p 2 z 0 ) + 1 = ) [ γ 1 + ωz0 ) 2γ 1 ωz 0 ) If x > 0 then [pz arg z0 p ) α ] z 0 ) 0 ) p 2 z 0 ) + 1 = = ix) γ [1 1 2 kγ1 + x2 )ix) 1 γ ] α. = γπ 2 + α arcctg ctg1 γ) π 2 2 ) 1 γ 1 + ωz0 ) kωz 0 ) 1 ωz 0 ) 1 ωz 0 )) ) kγ1 + x 2 )x 1+γ sin1 + γ) π 2 γπ 2 + α arcctg ctg1 γ) π ) = α + γ αγ) π 2 2. In a similar way, contradiction to condition 1) can be obtained for x < 0 and so, the assumption is wrong, i.e., ω < 1 for all z U. The sharpness of the result follows from the fact that for p = zp ) α p p = ) [ γ 1 + z 2γ 1 z 1+z 1 z ) γ, ) ] 1 γ α 1 + z z 1 z 1 z) ] α Putting p = zf f in Lemma 2 we obtain the main result of this paper. Theorem 1. M α α + γ αγ) S γ). This result is sharp. Remark 1. Sharpness of Theorem 1 means that for given 0 α 1 and 0 < γ 1, the number α + γ αγ is the biggest so the inclusion holds. Thus, Theorem 1 can be restated as Theorem 2. β α, γ) = α + γ αγ. Choosing γ = 1 in Theorem 1 we receive the same result as in [3]. Corollary 1. β α, 1) = 1, i.e., M α M α 1) S 1) S. Namely, let f A and Re [ zf ) α 1 + zf ) ] 1 α f f > 0, z U. Then f S and the result is sharp in the sense of Remark 1.

4 4 NIKOLA TUNESKI AND MASLINA DARUS By the definitions of classes M α β) and M α λ), having in mind that the disc ω 1 < sinβπ/2) lies in the angle arg ω < βπ/2, we receive the following corollary. Corollary 2. M α sin α + γ αγ)π/2)) S γ). This result is not sharp, i.e., λ α, γ) > sin α + γ αγ)π/2). In order to obtain the sharp result we give the next lemma that can be easily proved by imitating the proof of Lemma 2. Lemma 3. Let p be an analytic function in the unit disk U such that p0) = 1, and let 0 α 1 and 0 < γ 1. Also, let λ be the infimum of the function φx, k) = [1 ix)γ 1 ] α 2 kγ1 + x2 )ix) 1 γ 1, x R and k 1. Then zp ) α p p < λ, z U, implies arg p < γπ/2, z U. This result is sharp. For p = zf f in Lemma 3 we receive Theorem 3. Let λ be as defined in Lemma 3. Then M α λ) S γ) and the result is sharp, i.e., λ α, γ) = λ. Remark 2. All our attempts of finding explicit expression for λ α, γ) did not succeeded due to complicated calculations involved and it remains an open problem. Anyway, λ α, γ) can be calculated for separate values of γ as presented in the next corollary. 1, 0 α 1/2 Corollary 3. λ α, 1) = ) 2α 1,, i.e., if f A 1 + α 1/2 < α 1 and zf f 3α 2α 1 ) α 1 + zf f ) 1 α 1 < λ α, 1), z U, then f S and the result is sharp in the sense of Remark 1.

5 ON STRONG ALPHA-LOGARITHMICALLY CONVEX FUNCTIONS 5 Proof. According to Theorem 3, λ α, 1) is equal to the infimum of the function φx, k) = 1 + x k 2 + k ) 2α 2x 2, x R and k 1. Obviously, φx, k) is increasing by variable k and attains its infimum for k = 1. Further, the graph of φx, 1) is symmetric with respect to the y-axis, and so it is enough to study its behavior only for x 0. Since φ 1 xx, 1) = 2φ x x, 1) 3x α 3 3x ) 2α 2x 2 and x 0 it is easy to verify that: for α 1/2, φx, 1) is increasing function with minimal value φ0, 1) = 1; and for α > 1/2, function φx, 1) has minimal value ) 2α 1 3α 1 + α 2α 1 that is attained for x = 2α 1 3. References [1] P. L. Duren, Univalent functions, Springer-Verlag, New-York, [2] I. S. Jack, Functions starlike and convex of order α, J. London Math. Soc ), [3] D. K. Thomas, M. Darus, Alpha-logarithmically convex functions, Indian J. Pure Appl. Math , Faculty of Mechanical Engineering, Karpoš II b.b., 1000 Skopje, Republic of Macedonia address: nikolat@mf.ukim.edu.mk School of Mathematical Sciences, Faculty of Sciencs and Technology, University Kebangsaan Malaysia, Bangi Selangor, Malaysia address: maslina@pkrisc.cc.ukm.my