On neighborhoods of functions associated with conic domains

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)zf′(z)f(z)−(A−1)(B+1)zf′(z)f(z)−(A+1))>k|(B−1)zf′(z)f(z)−(A−1)(B+1)zf′(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)(zf′(z))′f′(z)−(A−1)(B+1)(zf′(z))′f′(z)−(A+1))>k|(B−1)(zf′(z))′f′(z)−(A−1)(B+1)(zf′(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].