On some sufficient conditions for univalence and starlikeness

Sokół and Nunokawa Journal of Inequalities and Applications 2012, 2012:282 http://www.journalofinequalitiesandapplications.com/content/2012/1/282 RESEARCH Open Access On some sufficient conditions for univalence and starlikeness Janusz Sokół1* and Mamoru Nunokawa2 * Correspondence: Department of Mathematics, Rzeszów University of Technology, Al. Powstańców Warszawy 12, Rzeszów, 35-959, Poland Full list of author information is available at the end of the article 1 Abstract In this work, the conditions for univalence, starlikeness and convexity are discussed. MSC: Primary 30C45; secondary 30C80 Keywords: strongly starlike functions; convex functions of order alpha; Jack’s lemma; Nunokawa’s lemma; Umezawa condition; univalence criteria 1 Introduction We shall consider the set H of all analytic functions in the open unit disc   D = z : |z| <  on the complex plane C and   A = f ∈ H : f (z) = z + a z + · · · . The class Sα* of starlike functions of order α <  may be defined as Sα* =   zf  (z) f ∈ A : Re > α, z ∈ D . f (z) The class Sα* and the class Kα of convex functions of order α <      zf  (z) > α, z ∈ D Kα := f ∈ A : Re  +  f (z)    * = f ∈ A : zf ∈ Sα were introduced by Robertson in []. If α ∈ [; ), then a function in either of these sets is univalent. In particular, we denote by S* = S * , K = K the classes of starlike and convex functions, respectively. 2 Preliminaries Lemma . Let O = , P = α + iaα, Q = x + iax and A ∈ (, +∞) be the points on the complex plane, where  < α ≤ A/, α < x and –∞ < a < ∞. Then we have |A – Q| |A – P| A – α < ≤ . |O – Q| |O – P| α (.) © 2012 Sokół and Nunokawa; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Sokół and Nunokawa Journal of Inequalities and Applications 2012, 2012:282 http://www.journalofinequalitiesandapplications.com/content/2012/1/282 Page 2 of 11 O = , P = α + iaα, A = , Q = x + iax,  < α ≤ A/, ∠PAO = ψ, ∠QOA = ϕ, ∠OAQ = ψ  . Figure 1 The lines and the points. Proof For a = , the assertion is obvious. For a = , consider the triangles OAP and OAQ, see Figure . Both of them have the same angle ϕ at the point O. Let the first have the angle ψ at the point A and the second have the angle ψ  at the point A. Then the hypothesis  < α ≤ A/ implies cos ϕ ≤ cos ψ. Further we have cos ϕ = α , |P| cos ψ = A–α . |A – P| This gives the second inequality of the assertion. The first one follows immediately from sin ψ  > sin ψ and |A – Q| sin ϕ = . |Q| sin ψ  |A – P| sin ϕ = , |P| sin ψ  3 Main result  n Theorem . Let p(z) =  + ∞ n=k≥ cn z be analytic in the unit disc D and α be a positive real number  < α ≤ /. Then suppose that there exists a point z , |z | <  such that   Re p(z) > α for |z| < |z | (.) and   Re p(z ) = p(z ) = α. (.) Then we have z p (z ) ≤ –k( – α). p(z ) Proof Let us put q(z) = p(z) –  , p(z) q() = . (.) Sokół and Nunokawa Journal of Inequalities and Applications 2012, 2012:282 http://www.journalofinequalitiesandapplications.com/content/2012/1/282 Page 3 of 11 Then the function q is analytic in |z| ≤ |z | <  and from the hypothesis of Theorem . and Lemma ., with A = , we have q(z) < –α α for |z| ≤ |z | and q(z ) = –α . α This shows that |q(z)| takes its maximum at z = z on the circle |z| = |z |. Then from FukuiSakaguchi [] and Jack’s [] lemmas, there exists a real number k ≥  such that z p (z ) z p (z ) z q (z ) = – q(z ) p(z ) –  p(z )     = z p (z ) – α– α = z p (z ) α(α – ) ≥ k. This shows that z p (z ) is a negative real number and z p (z ) z p (z ) = ≤ –k( – α). α p(z ) This completes the proof of Theorem ..  Theorem . is, in a certain sense, the supplement of Nunokawa’s lemma []. From Theorem . we have the following corollaries.  n Corollary . Let p(z) =  + ∞ n= cn z be analytic in the unit disc D and α be a positive real number  < α ≤ /. Suppose also that for arbitrary r,  < r < , p satisfies the condition   min Re p(z) = min p(z) |z|≤r |z|≤r (.) and   zp (z) Re p(z) + > α –  for |z| < . p(z) (.) Then we have   Re p(z) > α for |z| < . Proof If there exists a point z , |z | < , such that   Re p(z) > α for |z| < |z | (.) Sokół and Nunokawa Journal of Inequalities and Applications 2012, 2012:282 http://www.journalofinequalitiesandapplications.com/content/2012/1/282 Page 4 of 11 and   Re p(z ) = α,  < α ≤ /, then from the hypothesis of Corollary ., we have   Re p(z ) = p(z ) = α. Then from Theorem ., we have z p (z ) ≤ α – , p(z ) and therefore we have   z p (z ) Re p(z ) + ≤ α – . p(z ) This contradicts the hypothesis of Corollary . and it completes the proof of Corollary ..   n Corollary . Let f (z) = z + ∞ n= an z be analytic in the unit disc D and α be a positive real number  < α ≤ /. Suppose that for arbitrary r,  < r < , f satisfies the condition  zf  (z) zf  (z) = min |z|≤r f (z) f (z) (.)   zf  (z) > α –  for |z| < , Re  +  f (z) (.)  min Re |z|≤r and where – < α –  ≤ . Then we have    zf (z) >α Re f (z) for |z| < , (.) or f is starlike of order α. Proof Putting p(z) = zf  (z) , f (z) it follows that p(z) + zp (z) zf  (z) =+  . p(z) f (z) Then from Corollary ., we have (.).  Sokół and Nunokawa Journal of Inequalities and Applications 2012, 2012:282 http://www.journalofinequalitiesandapplications.com/content/2012/1/282 Page 5 of 11  n Theorem . Let p(z) =  + ∞ n=k≥ cn z be analytic in the unit disc D and α be a positive real number / < α < . Then suppose that there exists a point z ∈ D such that   Re p(z) > α for |z| < |z | (.) and   Re p(z ) = p(z ) = α. (.) Then we have   z p (z ) k( – α) z p (z ) = Re . ≤– p(z ) p(z )  (.) Proof Let us put q(z) =  – p(z) – , p(z) q() = . Then from Lemma ., with A = , we have that |q(z) + | takes its maximum value at z = z on the circle |z| = |z | or –α  – p(z) . = |z|=|z | p(z) α max q(z) +  = max |z|=|z | Applying Jack [], Miller-Mocanu [] and Fukui-Sakaguchi’s [] lemmas, there exists a real number m ≥ k such that z p (z ) z p (z ) z q (z ) =– – q(z )  – p(z ) p(z )     = –z p (z ) + –α α z p (z ) α( – α)   z p (z )  =– p(z )  – α =– ≥ m ≥ k. This shows that k( – α) z p (z ) ≤– . p(z )  This completes the proof of Theorem ..   n Corollary . Let f (z) = z + ∞ n= an z be analytic in the unit disc D and α be a positive real number / < α < . Suppose that for arbitrary r,  < r < , f satisfies the condition  min Re |z|≤r  zf  (z) zf  (z) = min |z|≤r f (z) f (z) (.) Sokół and Nunokawa Jo (...truncated)