On certain subclasses of close-to-convex and quasi-convex functions with respect to k-symmetric points

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1 J. Mah. Anal. Appl ) On cerain ubclae o cloe-o-convex and quai-convex uncion wih repec o -ymmeric poin Zhi-Gang Wang, Chun-Yi Gao, Shao-Mou Yuan College o Mahemaic and Compuing Science, Changha Univeriy o Science and Technology, Changha, Hunan 476, People epublic o China eceived 6 May 25 Available online 5 Sepember 25 Submied by H.M. Srivaava Abrac In he preen paper, he auhor inroduce wo new ubclae S ) φ) o cloe-o-convex uncion and C ) φ) o quai-convex uncion. The inegral repreenaion or uncion belonging o hee clae are provided, he convoluion condiion, growh heorem, diorion heorem and covering heorem or hee clae are alo provided. The reul obained generalize ome nown reul, and ome oher new reul are obained. 25 Elevier Inc. All righ reerved. Keyword: Cloe-o-convex uncion; Quai-convex uncion; Dierenial ubordinaion; Hadamard produc or convoluion); -Symmeric poin. Inroducion, deiniion and preliminarie Le A denoe he cla o uncion o he orm z)= z + a n z n,.) Thi wor i uppored by Scieniic eearch Fund o Hunan Provincial Educaion Deparmen, P China. * Correponding auhor. addre: zhigwang@63.com Z.-G. Wang) X/$ ee ron maer 25 Elevier Inc. All righ reerved. doi:.6/j.jmaa

2 98 Z.-G. Wang e al. / J. Mah. Anal. Appl ) 97 6 which are analyic in he open uni di U = z: z C and z <. Le S denoe he ubcla o A coniing o all uncion which are univalen in U. AloleP denoe he cla o uncion o he orm pz) = + p n z n z U), n= which aiy he condiion pz) >. We denoe by S, K, C and C he amiliar ubclae o A coniing o uncion which are, repecively, arlie, convex, cloe-o-convex and quai-convex in U. Thu, by deiniion, we have ee, or deail, [,2]; ee alo [3,4]) and S = : A and z z) z) K = : A and + z z) z) C = : A, g S, and C = : A, g K, and > z U), > z U), z z) gz) z z)) g z) > z U), > z U). Le z) and Fz) be analyic in U. Then we ay ha he uncion z) i ubordinae o Fz) in U, i here exi an analyic uncion ωz) in U uch ha ωz) z and z)= Fωz)), denoed by F or z) Fz).IFz)i univalen in U, hen he ubordinaion i equivalen o ) = F) and U) FU) ee [5]). A uncion z) A i in he cla S φ) i z z) z) φz) z U), where φz) P. The cla S φ) and a correponding convex cla Kφ) were deined by Ma and Minda [6]. And he reul abou he convex cla Kφ) can be eaily obained rom he correponding reul o uncion in S φ). Saaguchi [7] once inroduced a cla S o uncion arlie wih repec o ymmeric poin, which coni o uncion z) S aiying he inequaliy z z) > z U). z) z) Following him, many auhor dicued hi cla and i ubclae ee [8 6]). Moivaed by S, we can eaily obain he ollowing cla C o uncion convex wih repec o ymmeric poin. Deiniion. Le C denoe he cla o uncion in S aiying he inequaliy z z)) z) + > z U). z)

3 Z.-G. Wang e al. / J. Mah. Anal. Appl ) Now, we inroduce he ollowing clae o analyic uncion wih repec o -ymmeric poin and obain ome inereing reul. Deiniion 2. Le S ) φ) denoe he cla o uncion in S aiying he condiion z z) z) φz) z U), where φz) P, i a ixed poiive ineger and z) i deined by he ollowing equaliy: z) = ε ν ε ν z ) ε = )..2) ν= I = 2 and φz) = + z)/ z), hen he cla S ) φ) reduce o he cla S.I = 2, hen he cla S ) φ) reduce o he cla S φ), which wa inroduced and inveigaed recenly by avichandran [4]. I φz) = + βz)/ αβz) α, <β ), hen he cla S ) φ) reduce o he cla S ) [α, β], which wa conidered recenly by Gao and Zhou [5]. Deiniion 3. Le C ) φ) denoe he cla o uncion in S aiying he condiion z z)) z) φz) z U), where φz) P, i a ixed poiive ineger and z) i deined by equaliy.2). I = 2 and φz) = + z)/ z), hen he cla C ) φ) reduce o he cla C.I = 2, hen he cla C ) φ) reduce o he cla C φ), which wa alo inroduced and inveigaed recenly by avichandran [4]. In our propoed inveigaion o uncion in he clae S ) mae ue o he ollowing deiniion. φ) and C ) φ), we hall alo Deiniion 4 Hadamard produc or convoluion). Given wo uncion,g A, where z) i given by.) and gz) i deined by gz) = z + b n z n, he Hadamard produc or convoluion) g i deined a uual) by g)z) = z + a n b n z n = g )z). In he preen paper, ir we prove ha he clae S ) φ) and C ) φ) are ubclae o he cla o cloe-o-convex uncion and he cla o quai-convex uncion, repecively. Then we provide he inegral repreenaion or uncion belonging o hee clae. A la, we provide he convoluion condiion, growh heorem, diorion heorem and covering heorem or hee clae. The reul obained generalize ome nown reul and ome oher new reul are obained.

4 Z.-G. Wang e al. / J. Mah. Anal. Appl ) Inegral repreenaion Fir we give wo meaningul concluion abou he clae S ) Theorem. Le z) C ) φ), hen z) K S. φ) and C ) φ). Proo. Suppoe ha z) C ) φ), rom he deiniion o C ) φ) we can ge z z)) z) > z U) 2.) ince φz) >. Subiuing z by ε μ z in 2.) repecively μ =,, 2,..., ; ε = ), hen 2.) i alo rue, ha i, ε μ z) + ε μ z ε μ z) > z U; μ =,, 2,..., ). 2.2) εμ z) According o he deiniion o z) and ε =, we now εμ z) = z). Then inequaliy 2.2) become ε μ z) + ε μ z ε μ z) z) > z U). 2.3) Le μ =,, 2,..., in 2.3) repecively, and umming hem we can ge μ= ε μ z) + z μ= εμ ε μ z) z) > z U), or equivalenly, z) + z z) z) > z U), ha i z) K S. emar. From Theorem and inequaliy 2.), we now ha i z) C ) φ), hen z) i a quai-convex uncion. So C ) φ) i a ubcla o he cla C o quai-convex uncion. By applying imilar mehod a in Theorem, we have Theorem 2. Le z) S ) φ), hen z) S S. emar 2. From Theorem 2 and he deiniion o S ) φ), we now ha i z) S ) φ), hen z) i a cloe-o-convex uncion. So S ) φ) i a ubcla o he cla C o cloe-o-convex uncion. In paricular, i φz) = + βz)/ αβz), hen he ollowing reul o Gao and Zhou [5] are obained a pecial cae o Theorem 2. Corollary. Le z) S ) [α, β], hen z) S S.

5 Z.-G. Wang e al. / J. Mah. Anal. Appl ) 97 6 C ) Now, we give he inegral repreenaion o uncion belonging o he clae S ) φ). Theorem 3. Le z) C ) φ), hen we have z) = z exp ε μ ζ μ= φω)) φ) and d dζ, 2.4) where z) i deined by equaliy.2), ωz) i analyic in U and ω) =, ωz) <. Proo. Suppoe ha z) C ) φ), rom he deiniion o C ) φ) we have z z)) z) = φ ωz) ), 2.5) where ωz) i analyic in U and ω) =, ωz) <. Subiuing z by ε μ z in 2.5) repecively μ =,, 2,..., ; ε = ), we have ε μ z) + ε μ z ε μ z) = φ ω ε μ z )) μ =,, 2,..., ). 2.6) εμ z) I i eay o now ha εμ z) = z), umming 2.6) we can obain z z)) z) = φ ω ε μ z )), 2.7) μ= rom equaliy 2.7) we ge z z)) z z) z = μ= Inegraing equaliy 2.8) we have ha i, log z) = z μ= z) = exp μ= ε μ z φωε μ z)). 2.8) z φωε μ ζ)) ζ φω)) dζ = μ= Thereore, inegraing equaliy 2.9) we can obain equaliy 2.4). Theorem 4. Le z) C ) φ), hen we have z z)= ξ exp ξ ε μ ζ μ= ε μ z φω)) d, d. 2.9) φω)) where ωz) i analyic in U and ω) =, ωz) <. d φ ωζ) ) dζ dξ, 2.)

6 2 Z.-G. Wang e al. / J. Mah. Anal. Appl ) 97 6 Proo. Suppoe ha z) C ) φ); rom equaliie 2.5) and 2.9) we have z z) ) = z) φ ωz) ) = exp Inegraing equaliy 2.), we can obain z) = z z exp ε μ ζ μ= μ= φω)) ε μ z φω)) Thereore, inegraing equaliy 2.2) we can obain equaliy 2.). By applying imilar mehod a in Theorem 3, we have Theorem 5. Le z) S ) φ), hen we have z) = z exp μ= ε μ z φω)) d φ ωz) ). 2.) d φ ωζ) ) dζ. 2.2) d, where z) i deined by equaliy.2), ωz) i analyic in U and ω) =, ωz) <. Corollary 2. [5] Le z) S ) [α, β], hen we have z) = z exp ε μ z μ= + α)βω) αβω)) d where z) i deined by equaliy.2), ωz) i analyic in U and ω) =, ωz) <., By applying imilar mehod a in Theorem 4, we have Theorem 6. Le z) S ) φ), hen we have z)= z exp ε μ ζ μ= φω)) d φ ωζ) ) dζ, where ωz) i analyic in U and ω) =, ωz) <. Corollary 3. [5] Le z) S ) [α, β], hen we have z)= z exp ε μ ζ μ= + α)βω) αβω)) d where ωz) i analyic in U and ω) =, ωz) <. + βωζ) αβωζ ) dζ,

7 Z.-G. Wang e al. / J. Mah. Anal. Appl ) Convoluion condiion In hi ecion, we provide he convoluion condiion or he clae S ) φ) and C ) φ). Theorem 7. Le z) A and φz) P, hen z) S ) φ) i and only i [ z z z) 2 φ e iθ) )] hz) 3.) or all z U and θ<2π, where hz) i given by 3.6). Proo. Suppoe ha z) S ) φ), ince z z) z) φz) i and only i z z) z) φ e iθ) 3.2) or all z U and θ<2π. I i eay o now ha he condiion 3.2) can be wrien a z z) z)φ e iθ)). 3.3) z On he oher hand, i i well nown ha z z z) = z) z) ) And rom he deiniion o z), we now z) = z + a n c n z n = h)z), 3.5) where hz) = z + c n z n or, n= l +, c n =, n l +. Subiuing 3.4) and 3.5) ino 3.3), we can ge 3.). Thi complee he proo o Theorem 7. By applying imilar mehod a in Theorem 7, we have Theorem 8. Le z) A and φz) P, hen z) C ) φ) i and only i [ z z z z) 2 φ e iθ) ) ] hz) or all z U and θ<2π, where hz) i given by 3.6). 3.6)

8 4 Z.-G. Wang e al. / J. Mah. Anal. Appl ) Growh, diorion and covering heorem C ) Finally, we provide he growh, diorion and covering heorem or he clae S ) φ) and φ). For he purpoe o hi ecion, aume ha he uncion φz) i an analyic uncion wih poiive real par in he uni di U, φu) i convex and ymmeric wih repec o he real axi, φ) = and φ )>. The uncion φn z) n = 2, 3,...)deined by φn ) = φn ) = and + z φn z) φn z) = φ z n ) are imporan example o uncion in Kφ). The uncion h φn z) aiying h φn z) = z φn z) are example o uncion in S φ). Wrie φ2 z) imply a φ z) and h φ2 z) imply a h φ z). In order o prove our nex heorem, we hall require he ollowing lemma. Lemma. [7] Le z)= z + a + z + + Kφ), hen we have [ φ r )] / z) [ φ r )] /. Now we give he ollowing heorem. Theorem 9. Le min z =r φz) =φ r), max z =r φz) =φr), z =r<.iz) C ) φ), hen we have and r r r φ ) [ φ )] / d z) r r φ ) [ φ )] / d d z) U) ω: ω Thee reul are harp. φ) [ φ )] / d, 4.) r φ) [ φ )] / d d, 4.2) φ ) [ φ )] / d d. 4.3) Proo. Suppoe ha z) C ) φ), and φz) i convex and ymmeric wih repec o he real axi; i ollow ha z) = ε ν ε ν z ) = ε [ε ν ν z + a n ε ν z ) ] n = z + ν= ν= a l+ z l+ Kφ). l= Thu, by Lemma, we have [ φ r )] / z) [ φ r )] /.

9 Z.-G. Wang e al. / J. Mah. Anal. Appl ) Now, or z =r<, we have φ r) [ φ r )] / z z) ) = z z)) z) z) φr)[ φ r )] /. 4.4) By inegraing 4.4) rom o r, we can ge 4.). 4.2) ollow orm 4.). And 4.3) ollow orm 4.2), ince r φ ) [ φ )] / d d i increaing in, ) and bounded by. The reul are harp or he uncion z z)= φ ) [ φ ince i ha real coeicien and i in Kφ). )] / d d C ) φ), The proo o Theorem below i much ain o ha o Theorem 7 in [4], here we omi he deail. Theorem. Le min z =r φz) =φ r), max z =r φz) =φr), z =r<. Iz) S ) φ), hen we have h φ r) z) h φ r), h φ r) z) hφ r), and U) ω: ω h ). Thee reul are harp. Acnowledgmen The auhor are graeul o he reeree or hi careul reading and conrucive criicim o he original manucrip. eerence [] P.L. Duren, Univalen Funcion, Springer-Verlag, New Yor, 983. [2] H.M. Srivaava, S. Owa Ed.), Curren Topic in Analyic Funcion Theory, World Scieniic, Singapore, 992. [3] S. Owa, e al., Cloe-o-convexiy, arliene, and convexiy o cerain analyic uncion, Appl. Mah. Le. 5 22) [4] K.I. Noor, On quai-convex uncion and relaed opic, Inerna. J. Mah. Mah. Sci. 987) [5] C. Pommerene, Univalen Funcion, Vandenhoec and uprech, Göingen, 975. [6] W.C. Ma, D. Minda, A uniied reamen o ome pecial clae o univalen uncion, in: Proc. Con. Complex Analyi, Tianjin, 992, in: Con. Proc. Lecure Noe Anal., I, Inerna. Pre, Cambridge, MA, 994, pp [7] K. Saaguchi, On cerain univalen mapping, J. Mah. Soc. Japan 959) [8] J. Saniewicz, Some remar on uncion arlie wih repec o ymmeric poin, Ann. Univ. Mariae Curie- Słodowa Sec. A 9 97) [9] J. Thangamani, On arlie uncion wih repec o ymmeric poin, Indian J. Pure Appl. Mah. 98)

10 6 Z.-G. Wang e al. / J. Mah. Anal. Appl ) 97 6 [] H. Silverman, E.M. Silvia, Subclae o arlie uncion ubordinae o convex uncion, Canad. J. Mah ) []. Parvaham, S. adha, On α-arlie and α-cloe-o-convex uncion wih repec o n-ymmeric poin, Indian J. Pure Appl. Mah ) [2] T.N. Shanmugam, Convoluion and dierenial ubordinaion, Inerna. J. Mah. Mah. Sci ) [3] J. Soól, e al., On ome ubcla o arlie uncion wih repec o ymmeric poin, Zezyy Nau. Poliech. zezowiej Ma. Fiz. 2 99) [4] V. avichandran, Sarlie and convex uncion wih repec o conjugae poin, Aca Mah. Acad. Paedagog. Nyházi N.S.) 2 24) [5] C.-Y. Gao, S.-Q. Zhou, A new ubcla o cloe-o-convex uncion, Soochow J. Mah. 3 25) [6] T.V. Sudharan, e al., On uncion arlie wih repec o ymmeric and conjugae poin, Taiwanee J. Mah ) [7] I. Graham, D. Varolin, Bloch conan in one and everal variable, Paciic J. Mah )