Journal of the Ramanujan Mathematical Society
Volume 39, Issue 2, June 2024 pp. 143–161.
On the spectrum of complex unit gain graphs
Authors:
Molla Basir Ahamed and Vasudevarao Allu
Author institution:Department of Mathematics, Jadavpur University, Kolkata~700~032, West Bengal, India.
Summary:
The Bohr radius for the class of harmonic functions of the form f(z) = h + g
in the unit disk ⅅ := {z ∈ ℂ : |z| < 1}, where
h(z) = ⅀ {n=0} {∞} a{n}z{n} and g(z) = ⅀ {n=1} {∞}
b{n}z{n} is
the largest radius r{f}, 0 < r{f} < 1 such that
⅀ {n=1}{∞} (|a{n}| + |b{n}|)|z|{n} ≤ d(f(0),∂
f(ⅅ))
holds for |z| = r≤ r{f}, where
d(f(0),∂ f(ⅅ)) is the Euclidean distance between f(0)
and the boundary of f(ⅅ). In~this paper, we~prove two-type of
improved versions of the Bohr inequalities, one for certain class of harmonic
and univalent functions and the other is for stable harmonic mappings. It is
observed in the paper that to obtain sharp Bohr inequalities it is enough to
consider any non-negative real coefficients of the quantity S{r}/π. As~a
consequence of the main result, we~prove corollaries showing the precise
value of the sharp Bohr radius.
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