Sufficient Conditions for Strong Starlikeness
Tóm tắt
Let p be an analytic function defined on the open unit disc
$$\mathbb {D}$$
with
$$p(0)=1$$
and
$$0< \alpha \le 1$$
. The conditions on complex valued functions C, D, and E are obtained for p to be subordinate to
$$((1+z)/(1-z))^{\alpha }$$
when
$$C(z) z^{2}p''(z)+D(z)zp'(z) + E(z)p(z)=0$$
. Sufficient conditions for confluent (Kummer) hypergeometric function and generalized and normalized Bessel function of the first kind of complex order to be subordinate to
$$((1+z)/(1-z))^{\alpha }$$
are obtained as applications. The conditions on
$$\alpha $$
and
$$\beta $$
are derived for p to be subordinate to
$$((1+z)/(1-z))^{\alpha }$$
when
$$1+\beta zp'(z)/p^{n}(z)$$
with
$$n=1,2$$
is subordinate to
$$1+4z/3+2z^{2}/3=:\varphi _{CAR}(z)$$
. Similar problems were investigated for
$${{\,\mathrm{Re}\,}}p(z)>0$$
when the function
$$p(z)+\beta zp'(z)/p^{n}(z)$$
with
$$n=0,2$$
is subordinate to
$$\varphi _{CAR}(z)$$
. The condition on
$$\beta $$
is determined for p to be subordinate to
$$((1+z)/(1-z))^{\alpha }$$
when
$$p(z)+\beta zp'(z)/p^{n}(z)$$
with
$$n=0,1,2$$
is subordinate to
$$((1+z)/(1-z))^{\alpha }$$
.
Tài liệu tham khảo
Acharya, A.P.: Univalence criteria for analytic functions and applications to hypergeometric functions, Ph.D. Diss., University of Würzburg (1997)
Ali, R.M., Cho, N.E., Jain, N.K., Ravichandran, V.: Radii of starlikeness and convexity of functions defined by subordination with fixed second coefficients. Filomat 26, 553–561 (2012)
Ali, R.M., Cho, N.E., Ravichandran, V., Sivaprasad Kumar, S.: Differential subordination for functions associated with the lemniscate of Bernoulli. Taiwan. J. Math. 16(3), 1017–1026 (2012)
Ali, R.M., Mondal, S.R., Ravichandran, V.: On the Janowski convexity and starlikeness of the confluent hypergeometric function. Bull. Belg. Math. Soc. Simon Stevin 22(2), 227–250 (2015)
Baricz, Á.: Generalized Bessel Functions of the First Kind. Lecture Notes in Mathematics, vol. 1994. Springer, Berlin (2010)
Baricz, Á.: Applications of the admissible functions method for some differential equations. Pure Math. Appl. 13(4), 433–440 (2002)
Baricz, Á.: Geometric properties of generalized Bessel functions. Publ. Math. Debrecen 73(1–2), 155–178 (2008)
Bohra, N., Ravichandran, V.: On confluent hypergeometric functions and generalized Bessel functions. Anal. Math. 43(4), 533–545 (2017)
Cho, N.E., Kwon, O.S., Ravichandran, V.: Coefficient, distortion and growth inequalities for certain close-to-convex functions. J. Inequal. Appl. 2011, 100 (2011)
Goodman, A.W.: Univalent Functions, vol. II. Mariner, Tampa (1983)
Janowski, W.: Extremal problems for a family of functions with positive real part and for some related families. Ann. Polon. Math. 23, 159–177 (1970/1971)
Lee, S.K., Ravichandran, V., Supramaniam, S.: Close-to-convexity and starlikeness of analytic functions. Tamkang J. Math. 46(2), 111–119 (2015)
Ma, W.C., Minda, D.: A unified treatment of some special classes of univalent functions. In: Proceedings of the Conference on Complex Analysis (Tianjin, 1992), pp. 157–169. Conf. Proc. Lecture Notes Anal., vol. I. Int. Press, Cambridge (1992)
Miller, S.S., Mocanu, P.T.: Differential subordinations and inequalities in the complex plane. J. Differ. Equ. 67(2), 199–211 (1987)
Miller, S.S., Mocanu, P.T.: The theory and applications of second-order differential subordinations. Studia Univ. Babeş-Bolyai Math. 34(4), 3–33 (1989)
Miller, S.S., Mocanu, P.T.: Differential Subordinations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 225. Dekker, New York (2000)
Paprocki, E., Sokół, J.: The extremal problems in some subclass of strongly starlike functions. Zeszyty Nauk. Politech. Rzeszowskiej Mat. No. 20, 89–94 (1996)
Polatoğlu, Y., Bolcal, M.: Some radius problem for certain families of analytic functions. Turk. J. Math. 24(4), 401–412 (2000)
Ravichandran, V., Darus, M., Seenivasagan, N.: On a criteria for strong starlikeness. Aust. J. Math. Anal. Appl. 2(1), Art. 6, 1–12 (2005)
Ravichandran, V., Sharma, K.: Sufficient conditions for starlikeness. J. Korean Math. Soc. 52(4), 727–749 (2015)
Robertson, M.S.: Certain classes of starlike functions. Mich. Math. J. 32(2), 135–140 (1985)
Ruscheweyh, St., Singh, V.: On the order of starlikeness of hypergeometric functions. J. Math. Anal. Appl. 113(1), 1–11 (1986)
Shanmugam, T.N.: Convolution and differential subordination. Int. J. Math. Math. Sci. 12(2), 333–340 (1989)
Sharma, K., Jain, N.K., Ravichandran, V.: Starlike functions associated with a cardioid. Afr. Mat. (Springer) 27(5), 923–939 (2016)
Sharma, K., Ravichandran, V.: Applications of subordination theory to starlike functions. Bull. Iran. Math. Soc. 42(3), 761–777 (2016)
Sharma, K., Ravichandran, V.: Sufficient conditions for Janowski starlike functions. Stud. Univ. Babeş-Bolyai Math. 61(1), 63–76 (2016)
Sivaprasad Kumar, S., Kumar, V., Ravichandran, V., Cho, N.E.: Sufficient conditions for starlike functions associated with the lemniscate of Bernoulli. J. Inequal. Appl. 2013, Art. 176 (2013)
Sokół, J.: Radius problems in the class \(\cal{SL}\). Appl. Math. Comput. 214(2), 569–573 (2009)
Temme, N.M.: Special Functions. Wiley, New York (1996)