## On the convex combination of the Dziok-Srivastava operator.(English)Zbl 1124.33007

In 1999 Dziok and Srivastava have introduced the following operator $H_p(\alpha_1,\dots,\alpha_q; \beta_1,\dots,\beta_s): f(z) \mapsto h_p(\alpha_1,\dots,\alpha_q; \beta_1,\dots,\beta_s; z)\ast f(z)$ with $$h_p(\alpha_1,\dots,\alpha_q; \beta_1,\dots,\beta_s; z) = z^p\cdot _{q}F_{s}(\alpha_1,\ldots,\alpha_q; \beta_1,\dots,\beta_s; z)$$ and $$_{q}F_{s}$$ being the generalized hypergeometric function. This operator is known to be acting on the subspace $${\mathcal A}_p^k$$ of those analytic in the unit disc functions having the following representation $f(z) = z^p + \sum_{n=k}^{\infty} a_n z^n.$ Convex combinations of the Dziok-Srivastava operators are introduced and studied in the paper. They are proved to be acting on the subclasses $$V_p^k(s; A, B; t)$$ of those analytic functions having the form $f(z) = z^p - \sum_{n=k}^{\infty} a_n z^n \quad (a_n \geq 0)$ and satisfying the following condition $(1 - t) \frac{H_{p,s}(\alpha_1) f(z)}{z^p} + t \frac{H_{p,s}(\alpha_1 + 1) f(z)}{z^p} \prec \frac{1 + A z}{1 + B z},$ where $$H_{p,s} = H_p(\alpha_1,\dots,\alpha_{s+1}; \beta_1,\dots,\beta_s)$$. The set of the extreme points of these classes are characterized.

### MSC:

 33C20 Generalized hypergeometric series, $${}_pF_q$$ 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination 46B22 Radon-Nikodým, Kreĭn-Milman and related properties
Full Text:

### References:

 [1] Hallenbeck, D.; MacGregor, T.H., Linear problems and convexity techniques in geometric function theory, (1984), Pitman Advanced Publishing Program Boston, Pitman · Zbl 0581.30001 [2] Dziok, J.; Srivastava, H.M., Classes of analytic functions associated with the generalized hypergeometric function, Appl. math. comput., 103, 1-13, (1999) · Zbl 0937.30010 [3] Liu, J.-L.; Srivastava, H.M., Certain properties of the dziok – srivastava operator, Appl. math. comput., 159, 485-493, (2004) · Zbl 1081.30021 [4] Dziok, J.; Srivastava, H.M., Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function, Adv. stud. contemp. math., 5, 115-125, (2002) · Zbl 1038.30009 [5] Dziok, J.; Srivastava, H.M., Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral transform. spec. funct., 14, 7-18, (2003) · Zbl 1040.30003 [6] Gangadharan, A.; Shanmugam, T.N.; Srivastava, H.M., Generalized hypergeometric functions associated with k-uniformly convex functions, Comput. math. appl., 44, 1515-1526, (2002) · Zbl 1036.33003 [7] Kim, Y.C.; Srivastava, H.M., Fractional integral and other linear operators associated with the Gaussian haypergeometric function, Complex variables theor. appl., 34, 293-312, (1997) · Zbl 0951.30010 [8] Liu, J.-L., Strongly starlike functions associated with the dziok – srivastava operator, Tamkang J. math., 35, (2004) · Zbl 1064.30006 [9] Piejko, K.; Sokół, J., On the dziok – srivastava operator under multivalent analytic functions, Appl. math. comput., 177, 839-843, (2006) · Zbl 1100.30018 [10] Shanmugam, T.N.; Jeyaraman, M.P., Strongly mocanu convex functions associated with the dziok – srivastava operator, Far east J. math. sci., 20, 49-57, (2006) · Zbl 1095.30015 [11] Aghalary, R.; Azadi, Gh., The dziok – srivastava operator and k-uniformly starlike functions, J. inequal. pure appl. math., 6, 2, (2005), Article 52, 7pp · Zbl 1089.30009 [12] Sivaprasad Kumar, S.; Taneja, H.C.; Ravichandran, V., Classes of multivalent functions defined by dziok – srivastava linear operator and multiplier transformation, Kyungpook math. J., 46, 97-109, (2006) · Zbl 1104.30016 [13] Dziok, J., Classes of functions defined by certain differential-integral operator, J. comp. appl. math., 105, 245-255, (1999) · Zbl 0946.30007 [14] J. Dziok, On some applications of the Briot-Bouquet differential subordination, J. Math. Anal. Appl., in press. · Zbl 1158.30304 [15] Montel, P., Sur LES families de functions analytiques qui admettent des valeurs exceptionelles dans un domaine, Ann. sci. ecole norm. sup., 23, 487-535, (1912) · JFM 43.0509.05 [16] Krein, M.; Milman, D., On the extreme points of regularly convex sets, Stud. math., 9, 133-138, (1940) · JFM 66.0533.01 [17] Royden, H.L., Real analysis, (1988), Prentice-Hall Englewood Cliffs, NJ · Zbl 0704.26006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.