Classes of meromorphic harmonic functions and duality principle
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Classes of meromorphic harmonic functions and duality principle Jacek Dziok1 Received: 22 July 2020 / Revised: 22 July 2020 / Accepted: 8 October 2020 © The Author(s) 2020
Abstract We introduce new classes of meromorphic harmonic univalent functions. Using the duality principle, we obtain the duals of such classes of functions leading to coefficient bounds, extreme points and some applications for these functions. Keywords Subordination · Duality · Extreme points · Meromorphic harmonic functions · Varying coefficients Mathematics Subject Classification Primary 30C45; Secondary 30C80
1 Introduction A continuous function f = u +iv is a complex-valued harmonic function in a domain D⊂C if both u and v are real harmonic in D. In any simply connected domain D, we write f = h + g, where h and g are analytic in D. A necessary and sufficient condition for f to be locally one-to-one and orientation preserving in D is that |g (z)| < |h (z)| for z ∈ D (see Clunie and Sheil-Small [4]). Functions that are harmonic and univalent in D = {z : |z| > 1} are investigated by Hengartner and Schober [7]. In particular, it was shown in [7] that a complex-valued, harmonic, orientation preserving univalent mapping f , defined on D and satisfying f (∞) = ∞, must admit the representation f (z) = h(z) + g(z) + A log |z|
(1)
∞ −n −n where h(z) = αz + ∞ n=0 an z , g(z) = βz + n=0 bn z , 0 ≤ |β| < |α|, and ω = f z / f z is analytic and satisfies |ω(z)| < 1 for z ∈ D. We remove the logarithmic singularity in (1) by letting A = 0 and also let α = 1 and β = 0 and focus on the family H of meromorphic harmonic orientation preserving univalent mappings of the form
B 1
Jacek Dziok [email protected] Institute of Mathematics, University of Rzeszów, ul. Prof. Pigonia 1, 35-310 Rzeszow, Poland 0123456789().: V,-vol
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J. Dziok
f (z) = h(z) + g(z) where h(z) = z +
∞ n=1
an z −n , g(z) =
∞ n=1
bn z −n .
(2)
We say that a function f ∈ H is harmonic starlike in D if ∂ arg f r eit ≥ 0 ∂t or Re
DH f (z) ≥0 f (z)
where z = r eit ∈D, 0≤t≤2π , and DH f (z) = zh (z) − zg (z) = z −
∞ n an z −n − bn z −n . n=1
For l = 1, 2 and functions fl ∈ H of the form fl (z) = z +
∞ al,n z −n + bl,n z −n
(3)
n=1
we definethe convolution of f 1 and f 2 by ( f 1 ∗ f 2 ) (z) = f 1 (z) ∗ f 2 (z) = z +
∞ a1,n a2,n z −n + b1,n b2,n z −n . n=1
For m ∈ N0 := {0} ∪N = {1, 2, ...} and f = h + g∈H we define the linear operator m : → by DH H H m m := DH f (z) = Dm h(z) + (−1)m Dm g(z) DH ∞ m n an z −n + (−1)m bn z −n =z+ n=1
where D h(z) = h(z) ∗ z + m
m 1 = 1, m n =
(−1)m z(1 − 1z )m+1
= h(z) ∗ z +
(m + 1) · . . . · (m + n − 1) ; (n = 2, 3, ...) . (n − 1)!
0 f = f and D 1 f = D f . We note that DH H H
(−z)m (z − 1)m+1
,
Classes of meromorphic harmonic functions and duality…
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In 2008 Mauir [13] considered a weak subordination for complex-valued harmonic functions defined in the open unit disk := {z : |z| < 1}. In [8] Jahangiri (see also [10]) investigated the classes
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