On neighborhoods of functions associated with conic domains

Abstract

Abstract Let k − ST [A, B], k ≥ 0, −1 ≤ B < A ≤ 1 be the class of normalized analytic functions defined in the open unit disk satisfying ℜ ( ( B − 1 ) z f ′ ( z ) f ( z ) − ( A − 1 ) ( B + 1 ) z f ′ ( z ) f ( z ) − ( A + 1 ) ) > k | ( B − 1 ) z f ′ ( z ) f ( z ) − ( A − 1 ) ( B + 1 ) z f ′ ( z ) f ( z ) − ( A + 1 ) − 1 | . $$\Re \left( {{{(B - 1){{zf'(z)} \over {f(z)}} - (A - 1)} \over {(B + 1){{zf'(z)} \over {f(z)}} - (A + 1)}}} \right) > k\left| {{{(B - 1){{zf'(z)} \over {f(z)}} - (A - 1)} \over {(B + 1){{zf'(z)} \over {f(z)}} - (A + 1)}} - 1} \right|.$$ and let k − UCV [A, B], k ≥ 0, −1 ≤ B < A ≤ 1 be the corresponding class satisfying ℜ ( ( B − 1 ) ( z f ′ ( z ) ) ′ f ′ ( z ) − ( A − 1 ) ( B + 1 ) ( z f ′ ( z ) ) ′ f ′ ( z ) − ( A + 1 ) ) > k | ( B − 1 ) ( z f ′ ( z ) ) ′ f ′ ( z ) − ( A − 1 ) ( B + 1 ) ( z f ′ ( z ) ) ′ f ′ ( z ) − ( A + 1 ) − 1 | . $$\Re \left( {{{(B - 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A - 1)} \over {(B + 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A + 1)}}} \right) > k\left| {{{(B - 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A - 1)} \over {(B + 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A + 1)}} - 1} \right|.$$ For an appropriate δ > 0, the δ neighborhood of a function f ∈ k − UCV [A, B] is shown to consist of functions in the class k − ST [A, B].