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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.

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

[28]

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

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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.

[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.

[6]

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

[7]

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

[8]

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

[9]

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

[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.

[11]

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

[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.

[13]

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

[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.

[15]

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

[16]

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

[17]

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

[18]

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

[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.

[20]

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

[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.

[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.

[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.

[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.

[25]

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

[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.

[27]

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

[28]

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

[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.

[30]

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

[31]

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

[32]

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

[33]

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

[34]

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

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