Research Article Sufficient Conditions and Applications for Carathéodory Functions

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1 Mathematics Volume 2013, Article ID , 4 pages Research Article Sufficient Conditions and Applications for Carathéodory Functions Neng Xu School of Mathematics and Statistics, Changshu Institute of Technology, Changshu, Jiangsu , China Correspondence should be addressed to Neng Xu; xun@cslg.edu.cn Received 27 November 2012; Accepted 19 December 2012 Academic Editor: Jen-Chih Yao Copyright 2013 Neng Xu. is 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. Let PP betheclassoffunctionsoftheformpppppp p p p p nnnn pp nn zznn which are analytic in UU. By using the method of differential subordinations, we give some sufficient conditions for Carathéodory function, that is, if pppppp p pp satis es pppppp p p pp p pppp p pp and pppppp mm + λ (zzzzzzzzzzzz μμ ) h(zzzz where h(zzz z zzz z zzzzzzzz z zzzzzz mm + λ λ λ λ 2 μμ (1 + aaaaa μμ, 1 bbbbbbbb bbbb, mm, μμ μ μμμ μμ μμ, mmmmmmm, then pppppp p pp p pppppppp p ppppp. Some useful consequences of this result are also given. 1. Introduction Let AA betheclassoffunctionsofthefollowingform: ff (zz) =zzzaa nn zz nn, (1) which are analytic in UU U UUU U UUUU U UU. We denote by SS the subclass of AA consisting of univalent functions and by SS and KK the usual subclasses of SS whose members are starlike (with respect to the origin) and close-to-convex, respectively. Finally, we denote by RR the family of functions fffffwhich satisfy the condition Re fffffff f f fff f fff. Itiswellknown that RRRRR. Let PP betheclassoffunctionsofthefollowingform: nnnn pp (zz) =1+ pp nn zz nn, (2) which are analytic in UU. If pppppp p pp satis es Re pppppp p 0 (zz z zzz, then we say that pppppp is the Carathéodory function. For Carathéodory functions, Nunokawa et al. [1] have shown some sufficient conditions applying the differential inequalities. In the present paper, using the method of differential subordinations, we derive certain conditions under which we have pppppp p pp p pppppppp p ppppp, where 1 bbbbbbb. Our results generalize or improve some results due to [1 4]. To prove our results, we need the following lemma due to Miller and Mocanu [5]. nnnn Lemma 1. Let gggggg be analytic and univalent in U and let θθθθθθ and φφφφφφ be analytic in a domain D containing gggggg, with φφφφφφ φ φ when ww w wwwwww. Set = zzzz (zz) φφ gg (zz), and suppose that (i) QQQQQQ is starlike univalent in UU, and h(zz) =θθgg (zz) +QQ(zz) (3) (ii) Re(zzz (zzzzzzzzzzz z zzzzz (gggggggggggggggggg g gggg (zzzz QQQQQQQ Q Q QQQ Q QQQ. If pppppp is analytic in UU, with ppppp p ppppp, pppppp p pp and θθ pp (zz) + zzzz (zz) φφ pp (zz) θθgg (zz) + zzzz (zz) φφ gg (zz) =h(zz), then pppppp p pppppp and gggggg is the best dominant of (4). 2. Main Results eorem 2. Let 1 bbbbbbb,, mmm mm m mmm mm mm, and mmmmmmm. If pppppp p pp satis es pppppp p p pp p pppp p pp and (4) pp(zz) mm + zzzz (zz) pp (zz) μμ h(zz), (5)

2 2 Mathematics where h (zz) = 1 + aaaa 1+bbbb mm + (aaaaa) zz (1 + bbbb) 2 μμ, μμ (6) (1 + aaaa) then pppppp p pp p pppppppp p ppppp and (1 + aaaaaaaa a aaaaa is the best dominant of (5). Proof. Let 1 bbbbbbb,, mmm mm m mmm mm mm, and choose DDD CC CCC{0} μμ μ μμ μμ μ μμ μμ (7) gg (zz) = 1+aaaa 1+bbbb, θθ(ww) =wwmm, φφ(ww) =. (8) wwμμ en gggggg is analytic and univalent in UU, ggggg g ggggg g g, pppppp p pp, θθθθθθ and φφφφφφ satisfy the conditions of the lemma. e function = zzzz (zz) φφ gg (zz) = is univalent and starlike in UU because Re zzzz (zz) Further, we have =1 2 μμ Re = μμ Re > μμ (aaaaa) zz (1+bbbb) 2 μμ (1+aaaa) μμ (9) bbbb aaaa μμ μμ 1+bbbb 1+aaaa 1 1+bbbb + μμ μμ 1 1+aaaa 1 1+ bb +μμ 1 0 (zz zzz). 1+ aa (10) θθ gg (zz) +QQ(zz) = 1+aaaa mm 1+bbbb (aaaaa) zz + (1+bbbb) 2 μμ (1+aaaa) μμ =h(zz), zzz (zz) = mm mmmmmmm 1+aaaa 1+bbbb + zzzz (zz) Since mmmmmmm, from (11)itiseasytoknowthat (zz zzz). (11) Re zzz (zz) >0 (zz zzz). (12) Hence the function h(zzz is close-to-convex and univalent in UU.Nowitfollowsfrom(5) (12) that θθ pp (zz) + zzzz (zz) φφ pp (zz) θθgg (zz) + zzzz (zz) φφ gg (zz) =h(zz). (13) erefore, by virtue of the lemma, we conclude that pppppp p gggggg and gggggg is the best dominant of (5). eproofofthe theorem is complete. Making use of the theorem, we can obtain a number of interesting results. Corollary 3. Let ββββ, αααα. If pppppp p pp satis es pp (zz) 2 + ββ αα zzzz (zz) then Re pppppp p p. αααα1+ββ zzzzzzz2 αα(1 zz) 2 =h 1 (zz) (zzzzz), (14) Proof. Let aaaa, bb b bb, λ λ, ββββ, αααα, mmmm, and μμμμinthetheorem,thenwehave gg (zz) = 1+zz 1 zz, θθ(ww) =ww2, φφ (ww) = ββ αα, QQ(zz) = 2ββββ αα(1 zz) 2 satisfy the conditions of the lemma. Note that Re zzz 1 (zz) = 2αα ββ (15) +1 Re 1+zz 1 zz >0 (zz zzz). (16) Hence, similar to the proof of the theorem, we conclude that pppppp p pppppp, that is, Re pppppp p p. Remark 4. Note that Corollary 3 was also proved by Nunokawa et al. [1] using another method. Taking aa a a, bb b bb, λ λ, αα α α, ββ β β, and mmmmmmmin the theorem, we have the following. Corollary 5. Let αααα, ββββ, pppppp p pp satis es pppppp p p pp p zzz z zz, and αααα (zz) +ββ zzzz (zz) pp (zz) then Re pppppp p p. Remark 6. Note that the function αα 1+zz 1 zz + 2ββββ 1 zz 2 =h 2 (zz) (zzzzz), (17) wwww 2 (zz) =αα 1+zz 1 zz + 2ββββ 1 zz 2 ααααα αααα (18) maps UU onto the ww-plane slit along the half-lines Rewwww, Im ww w ββββββ β βββ and Re ww w w, Imwwwwββββββ β βββ. Hence Corollary 5 coincides with the result obtained by Nunokawa et al. [1, eorem 2] using another method. Letting ffffff f ff, pppppp p pp (zzz, ααααin Corollary 5, we have the following. Corollary 7. If ffffff f ff, ff (zzz z z zz z zzzz z zz, and +ββ zzzz (zz) iiii (zz zzz), (19) where ββββis real and bbb b ββββ β βββ, then ffffff f ff.

3 Mathematics 3 Remark 8. Lewandowski et al. [3] provedifffffff f ff, ff (zzz z z zz z zzzz z zz, ββββ, and Re +ββ zzzz (zz) >0 (zz zzz), (20) then ffffff f ff.we see that Corollary 7 improves this result. Corollary 9. Let ffffff f ff with ffffffff (zzz z z zz z zzzz z zz. If zzzz (zz) ff (zz) for some δδ δδ δ δδ δ δδ, then ffffff f ff and zz2 ff 2 (zz) (1 δδ) zz 1+(1 δδ) zz =h 3 (zz) (21) 1 <1 δδ (zz zzz). (22) Proof. Letting ffffff f ff, ffffffff (zzz z z zz z zzzz z zz, then pppppp p pp 2 ff (zzzzzz 2 (zzz z zz, zzzz (zz) pp (zz) zzzz (zz) = ff (zz). (23) Taking mm m m, μμ μ μμ μ μ and aa a a a aa aa a aa a aa, bbbbinthetheorem,itfollowsfrom(5), (6),and(21) that (zz 2 ff (zzzzzzzz 2 (zzzz z zz z z z zz zzz z zzz, which implies that ffffff f ff (see [6]). Remark 10. Frassin and Darus [2]haveshownthatifffffff f AA and zzzz (zz) ff (zz) ff (zz) < 1 δδ 2 δδ (zz zzz) (24) for some δδ δδ δ δδ δ δδ, then (zz 2 ff (zzzzzz 2 (zzzzzzz z zzzz zzz z UUU. For 0 δδδδ, the function h 3 (zz) = (1 δδ) zz 1 (1 δδ) zz is analytic and convex univalent in UU. Since (25) h 3 (UU) = ww wwww (1 δδ)2 δδ (2 δδ) < 1 δδ 0<δδδδ), δδ (2 δδ) h 3 (UU) = ww wwwwww 1 2 δδδδ), (26) the disk www w ww w wwwwww w www is properly contained in h 3 (UUU. erefore, in view of (21), we see that Corollary 9 is better thantheresultgivenin[2]. Letting ffffff f ff, pppppp p pp (zzz z zz, mm m m, μμ μ μ, aaaaaaaa, 0 αααα, and bb b bb in the theorem, we have the following. Corollary 11. If ffffff f ff,, 0 αααα, and + (zz) 1+(1 2αα) zz 1 zz 2 (1 αα) zz + (1 zz) 2 =h 4 (zz) (zzzzz), (27) then Re ff (zzz z zz. For the univalent function h 4 (zzz, wenow ndtheimage h 4 (UUU of the unit disk UU. Let h 4 (ee iiii ) = uu u uuuu, where uu and vv arereal.wehave uuuuuu Elimination of θθ yields (1 αα) 1 cos θθ, vvv(1 αα) sin θθ 1 cos θθ. (28) vv 2 = 2αα (1 αα) (1 αα)2 erefore, we conclude that h 4 (UU) = ww wwwwwwwwwww 2 > 2 (1 αα) uuu (1 αα) 2 2 (1 αα) uuu (29) αα. (30) Remark 12. Nunokawa and Hoshino [4] have proved that if ffffff f ff, and Re + (zz) >, (zz zzz), (31) 2 then Re ff (zzz z z. For αααα, h 4 (UUU properly contains the half plane Re ww w wwwww. Hence Corollary 11 with ααααis better thantheresultgivenin[4]. Acknowledgments is work was partially supported by the National Natural Science Foundation of China (Grant no ). e author would like to thank the referees for their careful reading and for making some valuable comments which have essentially improved the presentation of this paper. References [1] M. Nunokawa, S. Owa, N. Takahashi, and H. Saitoh, Sufficient conditions for Carathéodory functions, Indian Pure and Applied Mathematics, vol. 33, no. 9, pp , [2] B. A. Frasin and M. Darus, On certain analytic univalent functions, International Mathematics and Mathematical Sciences, vol. 25, no. 5, pp , [3] Z. Lewandowski, S. Miller, and E. Złotkiewicz, Generating functions for some classes of univalent functions, Proceedings of the American Mathematical Society, vol. 56, pp , [4] M. Nunokawa and S. Hoshino, One criterion for multivalent functions, Proceedings of the Japan Academy. A, vol. 67, no. 2, pp , 1991.

4 4 Mathematics [5] S. S. Miller and.. Mocanu, On some classes of rst order differential subordinations, e Michigan Mathematical Journal, vol. 32, no. 2, pp , [6] S. Ozaki and M. Nunokawa, e Schwarzian derivative and univalent functions, Proceedings of the American Mathematical Society, vol. 33, pp , 1972.

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