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December  2015, 8(6): 1277-1290. doi: 10.3934/dcdss.2015.8.1277

Higher order uniformly close-to-convex functions

1. 

COMSATS Institute of Information Technology, Department of Mathematics, Islamabad, Pakistan, Pakistan

Received  May 2015 Revised  September 2015 Published  December 2015

In this paper, we define and study some subclasses of analytic functions related with $k$-uniformly close-to-convex functions of higher order in the unit disc. These classes unify a number of classes previously studied. The results obtained include rate of growth of coefficients, inclusion relations, radius problems and necessary conditions for univalency. We derive many known results as special cases.
Citation: Khalida Inayat Noor, Muhammad Aslam Noor. Higher order uniformly close-to-convex functions. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1277-1290. doi: 10.3934/dcdss.2015.8.1277
References:
[1]

M. Acu, On a subclass of n-uniformly close-to-convex functions,, Gen. Math., 14 (2006), 55. Google Scholar

[2]

S. D. Bernardi, Convex and starlike univalent functions,, Trans. Amer. Math. Soc., 135 (1969), 429. doi: 10.1090/S0002-9947-1969-0232920-2. Google Scholar

[3]

D. A. Brannan, On function of bounded boundary rotation,, Proc. Edin. Math. Soc., 2 (): 339. Google Scholar

[4]

M. Caglar, H. Ohan and E. Deniz, Majorization for certain subclass of analytic functions involving the generalized Noor integral operator,, Filomat, 27 (2013), 143. doi: 10.2298/FIL1301143C. Google Scholar

[5]

N. E. Cho, S. Kwon and H. M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators,, J. Math. Anal. Appl., 292 (2004), 470. doi: 10.1016/j.jmaa.2003.12.026. Google Scholar

[6]

E. Denz, Univalence criteria for a general integral operator,, Filomat, 28 (2014), 11. doi: 10.2298/FIL1401011D. Google Scholar

[7]

A. W. Goodman, On uniformly starlike functions,, J. Math. Anal. Appl., 155 (1991), 364. doi: 10.1016/0022-247X(91)90006-L. Google Scholar

[8]

A. W. Goodman, On close-to-convex functions of higher order,, Ann. Univ. Budapest, 15 (1972), 17. Google Scholar

[9]

A. W. Goodman, Univalent Functions, Vol I, II,, Polygonal Publishing House, (1983). Google Scholar

[10]

I. B. Jung, Y. C. Kim and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operator,, J. Math. Anal. Appl., 176 (1993), 138. doi: 10.1006/jmaa.1993.1204. Google Scholar

[11]

S. Kanas and A. Wisniowska, Conic domains and starlike functions,, Rev. Roumaine Math. Pures. Appl., 45 (2000), 647. Google Scholar

[12]

S. Kanas, Techniques of the differential subordination for domain bounded by conic sections,, Int. J. Math. Math. Sci., 38 (2003), 2389. doi: 10.1155/S0161171203302212. Google Scholar

[13]

W. Kaplan, Close-to-convex schlicht functions,, Michigan J. Math., 1 (1952), 169. doi: 10.1307/mmj/1028988895. Google Scholar

[14]

R. J. Libera, Some classes of regular univalent functions,, Proc. Amer. Math. Soc., 16 (1965), 755. doi: 10.1090/S0002-9939-1965-0178131-2. Google Scholar

[15]

S. S. Miller and P. T. Mocanu, Differential Subordinations,, Theory and Applications, (2000). Google Scholar

[16]

E. J. Moulis, Generalizations of Robertson functions,, Pacific J. Math., 81 (1979), 167. doi: 10.2140/pjm.1979.81.167. Google Scholar

[17]

K. I. Noor, On quasi-convex functions and related topics,, Inter. J. Math. Math. Sci., 10 (1987), 241. Google Scholar

[18]

K. I. Noor, On generalization of close-to-convexity,, Inter. J. Math. Math. Sci., 23 (1981), 217. Google Scholar

[19]

K. I. Noor, On generalization of uniformly convex and related functions,, Comput. Math. Appl., 61 (2011), 117. doi: 10.1016/j.camwa.2010.10.038. Google Scholar

[20]

K. I. Noor, Higher order close-to-convex functions,, Math. Japonica, 37 (1992), 1. Google Scholar

[21]

K. I. Noor, R. Fayyaz and M. A. Noor, Some classes of k-uniformly functions with bounded radius rotation,, Appl. Math. Inform. Sci., 8 (2014), 527. doi: 10.12785/amis/080210. Google Scholar

[22]

K. I. Noor, W. Ul-Haq, M. Arif and S. Mustafa, On functions of bounded boundary and bounded radius rotations,, J. Inequa. Appl., (2009). doi: 10.1155/2009/813687. Google Scholar

[23]

K. I. Noor, M. Arif and W. Ul-Haq, On k-uniformly close-to-convex functions of complex order,, Appl. Math. Comput., 215 (2009), 629. doi: 10.1016/j.amc.2009.05.050. Google Scholar

[24]

K. I. Noor and N. Khan, Some Classes of $p$-valent analytic functions associated with hypergeometric functions,, Filomat, 29 (2015), 1031. doi: 10.2298/FIL1505031N. Google Scholar

[25]

K. I. Noor, N. Khan and M. A. Noor, On generalized spiral-like analytic functions,, Filomat, 28 (2014), 1493. doi: 10.2298/FIL1407493N. Google Scholar

[26]

K. I. Noor and M. A. Noor, Higher-order close-to-convex functions related with conic domain,, Appl. Math. Inform. Sci., 8 (2014), 2455. doi: 10.12785/amis/080541. Google Scholar

[27]

K. I. Noor and D. K. Thomas, Quasi-convex univalent functions,, Int. J. Math. Math. Sci., 3 (1980), 255. doi: 10.1155/S016117128000018X. Google Scholar

[28]

M. Obradovic and P. Ponnusanny, Radius of univalence of certain class of analytic functions,, Filomat, 27 (2013), 1085. doi: 10.2298/FIL1306085O. Google Scholar

[29]

T. O. Opoola and K. O. Babalola, Some applications of a lemma concerning analytic functions with positive real parts in the unit disk,, Int. J. Math. Comput. Sci., 2 (2007), 361. Google Scholar

[30]

R. Parvatham and S. Radha, On certain classes of analytic functions,, Ann. Polon Math., 49 (1988), 31. Google Scholar

[31]

B. Pinchuk, Functions with bounded boundary rotation,, Isr. J. Math., 10 (1971), 6. doi: 10.1007/BF02771515. Google Scholar

[32]

Ch. Pommerenke, On close-to-convex analytic functions,, Trans. Amer. Math. Soc., 114 (1965), 176. doi: 10.1090/S0002-9947-1965-0174720-4. Google Scholar

[33]

G. S. Salagean, Subclasses of univalent functions,, in Complex Analysis - Fifth Romanian-Finnish Seminar, (1013), 362. doi: 10.1007/BFb0066543. Google Scholar

[34]

D. K. Thomas, On Bazilevic functions,, Trans. Amer. Math. Soc., 132 (1968), 353. Google Scholar

show all references

References:
[1]

M. Acu, On a subclass of n-uniformly close-to-convex functions,, Gen. Math., 14 (2006), 55. Google Scholar

[2]

S. D. Bernardi, Convex and starlike univalent functions,, Trans. Amer. Math. Soc., 135 (1969), 429. doi: 10.1090/S0002-9947-1969-0232920-2. Google Scholar

[3]

D. A. Brannan, On function of bounded boundary rotation,, Proc. Edin. Math. Soc., 2 (): 339. Google Scholar

[4]

M. Caglar, H. Ohan and E. Deniz, Majorization for certain subclass of analytic functions involving the generalized Noor integral operator,, Filomat, 27 (2013), 143. doi: 10.2298/FIL1301143C. Google Scholar

[5]

N. E. Cho, S. Kwon and H. M. Srivastava, Inclusion relationships and argument properties for certain subclasses of multivalent functions associated with a family of linear operators,, J. Math. Anal. Appl., 292 (2004), 470. doi: 10.1016/j.jmaa.2003.12.026. Google Scholar

[6]

E. Denz, Univalence criteria for a general integral operator,, Filomat, 28 (2014), 11. doi: 10.2298/FIL1401011D. Google Scholar

[7]

A. W. Goodman, On uniformly starlike functions,, J. Math. Anal. Appl., 155 (1991), 364. doi: 10.1016/0022-247X(91)90006-L. Google Scholar

[8]

A. W. Goodman, On close-to-convex functions of higher order,, Ann. Univ. Budapest, 15 (1972), 17. Google Scholar

[9]

A. W. Goodman, Univalent Functions, Vol I, II,, Polygonal Publishing House, (1983). Google Scholar

[10]

I. B. Jung, Y. C. Kim and H. M. Srivastava, The Hardy space of analytic functions associated with certain one-parameter families of integral operator,, J. Math. Anal. Appl., 176 (1993), 138. doi: 10.1006/jmaa.1993.1204. Google Scholar

[11]

S. Kanas and A. Wisniowska, Conic domains and starlike functions,, Rev. Roumaine Math. Pures. Appl., 45 (2000), 647. Google Scholar

[12]

S. Kanas, Techniques of the differential subordination for domain bounded by conic sections,, Int. J. Math. Math. Sci., 38 (2003), 2389. doi: 10.1155/S0161171203302212. Google Scholar

[13]

W. Kaplan, Close-to-convex schlicht functions,, Michigan J. Math., 1 (1952), 169. doi: 10.1307/mmj/1028988895. Google Scholar

[14]

R. J. Libera, Some classes of regular univalent functions,, Proc. Amer. Math. Soc., 16 (1965), 755. doi: 10.1090/S0002-9939-1965-0178131-2. Google Scholar

[15]

S. S. Miller and P. T. Mocanu, Differential Subordinations,, Theory and Applications, (2000). Google Scholar

[16]

E. J. Moulis, Generalizations of Robertson functions,, Pacific J. Math., 81 (1979), 167. doi: 10.2140/pjm.1979.81.167. Google Scholar

[17]

K. I. Noor, On quasi-convex functions and related topics,, Inter. J. Math. Math. Sci., 10 (1987), 241. Google Scholar

[18]

K. I. Noor, On generalization of close-to-convexity,, Inter. J. Math. Math. Sci., 23 (1981), 217. Google Scholar

[19]

K. I. Noor, On generalization of uniformly convex and related functions,, Comput. Math. Appl., 61 (2011), 117. doi: 10.1016/j.camwa.2010.10.038. Google Scholar

[20]

K. I. Noor, Higher order close-to-convex functions,, Math. Japonica, 37 (1992), 1. Google Scholar

[21]

K. I. Noor, R. Fayyaz and M. A. Noor, Some classes of k-uniformly functions with bounded radius rotation,, Appl. Math. Inform. Sci., 8 (2014), 527. doi: 10.12785/amis/080210. Google Scholar

[22]

K. I. Noor, W. Ul-Haq, M. Arif and S. Mustafa, On functions of bounded boundary and bounded radius rotations,, J. Inequa. Appl., (2009). doi: 10.1155/2009/813687. Google Scholar

[23]

K. I. Noor, M. Arif and W. Ul-Haq, On k-uniformly close-to-convex functions of complex order,, Appl. Math. Comput., 215 (2009), 629. doi: 10.1016/j.amc.2009.05.050. Google Scholar

[24]

K. I. Noor and N. Khan, Some Classes of $p$-valent analytic functions associated with hypergeometric functions,, Filomat, 29 (2015), 1031. doi: 10.2298/FIL1505031N. Google Scholar

[25]

K. I. Noor, N. Khan and M. A. Noor, On generalized spiral-like analytic functions,, Filomat, 28 (2014), 1493. doi: 10.2298/FIL1407493N. Google Scholar

[26]

K. I. Noor and M. A. Noor, Higher-order close-to-convex functions related with conic domain,, Appl. Math. Inform. Sci., 8 (2014), 2455. doi: 10.12785/amis/080541. Google Scholar

[27]

K. I. Noor and D. K. Thomas, Quasi-convex univalent functions,, Int. J. Math. Math. Sci., 3 (1980), 255. doi: 10.1155/S016117128000018X. Google Scholar

[28]

M. Obradovic and P. Ponnusanny, Radius of univalence of certain class of analytic functions,, Filomat, 27 (2013), 1085. doi: 10.2298/FIL1306085O. Google Scholar

[29]

T. O. Opoola and K. O. Babalola, Some applications of a lemma concerning analytic functions with positive real parts in the unit disk,, Int. J. Math. Comput. Sci., 2 (2007), 361. Google Scholar

[30]

R. Parvatham and S. Radha, On certain classes of analytic functions,, Ann. Polon Math., 49 (1988), 31. Google Scholar

[31]

B. Pinchuk, Functions with bounded boundary rotation,, Isr. J. Math., 10 (1971), 6. doi: 10.1007/BF02771515. Google Scholar

[32]

Ch. Pommerenke, On close-to-convex analytic functions,, Trans. Amer. Math. Soc., 114 (1965), 176. doi: 10.1090/S0002-9947-1965-0174720-4. Google Scholar

[33]

G. S. Salagean, Subclasses of univalent functions,, in Complex Analysis - Fifth Romanian-Finnish Seminar, (1013), 362. doi: 10.1007/BFb0066543. Google Scholar

[34]

D. K. Thomas, On Bazilevic functions,, Trans. Amer. Math. Soc., 132 (1968), 353. Google Scholar

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