Mapping Properties of Hypergeometric Functions and Convolutions of Starlike or Convex Functions of Order alpha
نویسنده
چکیده
We determine the order of convexity of hypergeometric functions z 7→ F (a, b, c, z) as well as the order of starlikeness of shifted hypergeometric functions z 7→ zF (a, b, c, z), for certain ranges of the real parameters a, b and c. As a consequence we obtain the sharp lower bound for the order of convexity of the convolution (f ∗ g)(z) := ∑∞ n=0 anbnz n when f(z) = ∑∞ n=0 anz n is convex of order α ∈ [0, 1] and g(z) = ∑∞ n=0 bnz n is convex of order β ∈ [0, 1], and likewise we obtain the sharp lower bound for the order of starlikeness of f ∗ g when f, g are starlike of order α, β ∈ [1/2, 1], respectively. Further we obtain convexity in the direction of the imaginary axis for hypergeometric functions and for three ratios of hypergeometric functions as well as for the corresponding shifted expressions. In the proofs we use the continued fraction of Gauss, a theorem of Wall which yields a characterization of Hausdorff moment sequences by means of (continued) g-fractions, and results of Merkes, Wirths and Pólya. Finally we state a subordination problem.
منابع مشابه
A Subclass of Analytic Functions Associated with Hypergeometric Functions
In the present paper, we have established sufficient conditions for Gaus-sian hypergeometric functions to be in certain subclass of analytic univalent functions in the unit disc $mathcal{U}$. Furthermore, we investigate several mapping properties of Hohlov linear operator for this subclass and also examined an integral operator acting on hypergeometric functions.
متن کاملSome properties and results for certain subclasses of starlike and convex functions
In the present paper, we introduce and investigate some properties of two subclasses $ Lambda_{n}( lambda , beta ) $ and $ Lambda_{n}^{+}( lambda , beta ) $; meromorphic and starlike functions of order $ beta $. In particular, several inclusion relations, coefficient estimates, distortion theorems and covering theorems are proven here for each of these function classes.
متن کاملSome Properties of Certain Subclasses of Close-to-Convex and Quasi-convex Functions with Respect to 2k-Symmetric Conjugate Points
متن کامل
Applications of subordination theory to starlike functions
Let $p$ be an analytic function defined on the open unit disc $mathbb{D}$ with $p(0)=1.$ The conditions on $alpha$ and $beta$ are derived for $p(z)$ to be subordinate to $1+4z/3+2z^{2}/3=:varphi_{C}(z)$ when $(1-alpha)p(z)+alpha p^{2}(z)+beta zp'(z)/p(z)$ is subordinate to $e^{z}$. Similar problems were investigated for $p(z)$ to lie in a region bounded by lemniscate of Bernoulli $|w^{2}-1|=1$ ...
متن کاملPartial second-order subdifferentials of -prox-regular functions
Although prox-regular functions in general are nonconvex, they possess properties that one would expect to find in convex or lowerC2 functions. The class of prox-regular functions covers all convex functions, lower C2 functions and strongly amenable functions. At first, these functions have been identified in finite dimension using proximal subdifferential. Then, the definition of prox-regula...
متن کامل