On Meromorphic Harmonic Functions with Respect to -Symmetric Points

Theorem 2.1. Let f  h  g be of the form 1.3 and fk  hk  gk where hk and gk are given by 1.9. If ∞  n1 n −1k  1  α|an−1k1|  n −1k  1 −α|bn−1k1|Ω j mn, k  ∞  n2 n / lk1 nj1Cm, n|an|  |bn| ≤1 −α, 2.1

Theorem 2.2. Let fj  hj  gj, where hj and gj are given by 1.11, and fkj  hkj  gkj where hkj and gkj are given by 1.12. Then, fj ∈MHSk s j, m, α, if and only if the inequality 2.1 holds for the coefficient of fj  hj  gj and fkj  hkj  gkj.

Theorem 3.1. If fj  hj  gj ∈MHSk s j, m, α and 0 < |z|  r < 1, then 1 r − 1 −α 2jm  12 −αr ≤|fjz| ≤1 r  1 −α 2jm  12 −αr. 3.1

Theorem 3.1.

Corollary 3.2. If fj ∈MHSk s j, m, α, then fjU \ {0} ⊂  w : |w| < 2jm  12 −α −1 −α 2jm  12 −α  . 3.3

Theorem 3.3. Let fj  hj  gj where hj and gj are given by 1.11. Then, fj ∈MHSk s j, m, α if and only if fj,nz  ∞  n0 xnhjnz  yngjnz, 3.4 where hj,0  gj,0z  −1j/z, hj,nz  −1j/z  1 −α/njCm, nn  αΦnzn n  1, 2, 3, . . ., gj,nz  −1j/z−1j1−α/njCm, nn−αΦnzk n  1, 2, 3, . . ., ∞ n0xn  yn  1, xn ≥0, yn ≥0. In particular, the extreme points of MHSk s j, m, α are {hj,n} and {gj,n}.

Theorem 4.1. For 0 ≤β ≤α < 1, let fj ∈MHSk s j, m, α and Fj ∈MHSk s j, m, β. Then, fj∗Fj ∈MHSk s j, m, α ⊂MHSk s j, m, β.

Theorem 2.2. For Fj ∈MHSk s j, m, β, we note that |An| ≤1 and |Bn| ≤1. Now, for the convolution function fj∗Fj, we obtain ∞  n1 njCm, nn  βΦn 1 −β |an||An|  ∞  n1 njCm, nn −βΦn 1 −β |bn||Bn|

Theorem 4.2. Let the functions fj,t defined by 4.3 be in the class MHSk s j, m, α for every t  1, 2, . . . , ρ. Then, the functions ξtz defined by ξtz  ρ  t1 ctfjnz, 0 ≤ct ≤1, 4.4 are also in the class MHSk s j, m, α, where ρ t1ct  1.

Corollary 4.3. The class MHSk s j, m, α is close under convex linear combination.