<abstract xmlns="http://www.w3.org/1999/xhtml"><p>Let <italic>k</italic> − <italic>ST</italic> [<italic>A</italic>, <italic>B</italic>], <italic>k</italic> ≥ 0, −1 ≤ <italic>B</italic> &lt; <italic>A</italic> ≤ 1 be the class of normalized analytic functions defined in the open unit disk satisfying
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2015-0020_eq_g001.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>ℜ</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mo>></mo><mi>k</mi><mrow><mo>|</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>.</mo></math><tex-math>$$\Re \left( {{{(B - 1){{zf'(z)} \over {f(z)}} - (A - 1)} \over {(B + 1){{zf'(z)} \over {f(z)}} - (A + 1)}}} \right) > k\left| {{{(B - 1){{zf'(z)} \over {f(z)}} - (A - 1)} \over {(B + 1){{zf'(z)} \over {f(z)}} - (A + 1)}} - 1} \right|.$$</tex-math></alternatives></disp-formula>
and let <italic>k</italic> − <italic>UCV</italic> [<italic>A</italic>, <italic>B</italic>], <italic>k</italic> ≥ 0, −1 ≤ <italic>B</italic> &lt; <italic>A</italic> ≤ 1 be the corresponding class satisfying
<disp-formula><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2015-0020_eq_g002.png"></graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mi>ℜ</mi><mrow><mo>(</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>)</mo></mrow><mo>></mo><mi>k</mi><mrow><mo>|</mo><mrow><mfrac><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mo stretchy="false">(</mo><mi>A</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mrow><msup><mrow><mrow><mo>(</mo><mrow><mi>z</mi><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><mo>)</mo></mrow></mrow><mrow><mo>′</mo></mrow></msup></mrow><mrow><msup><mi>f</mi><mo>′</mo></msup><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mo>−</mo><mn>1</mn></mrow><mo>|</mo></mrow><mo>.</mo></math><tex-math>$$\Re \left( {{{(B - 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A - 1)} \over {(B + 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A + 1)}}} \right) > k\left| {{{(B - 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A - 1)} \over {(B + 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A + 1)}} - 1} \right|.$$</tex-math></alternatives></disp-formula>
For an appropriate <italic>δ</italic> > 0, the <italic>δ</italic> neighborhood of a function <italic>f</italic> ∈ <italic>k</italic> − <italic>UCV</italic> [<italic>A</italic>, <italic>B</italic>] is shown to consist of functions in the class <italic>k</italic> − <italic>ST</italic> [<italic>A</italic>, <italic>B</italic>].</p></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)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].

On neighborhoods of functions associated with conic domains

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

{"article-title":"On neighborhoods of functions associated with conic domains"}

Let k − ST [A, B], k ≥ 0, −1 ≤ B 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 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].

Let k − ST [A, B], k ≥ 0, −1 ≤ B < A ≤ 1 be the class of normalized analytic functions defined in the open unit disk...

On neighborhoods of functions associated with conic domains

Online veröffentlicht: 04. Apr. 2017

Seitenbereich: 291 - 302

Eingereicht: 08. Mai 2014

Akzeptiert: 30. Juni 2014

DOI: https://doi.org/10.1515/auom-2015-0020

Schlüsselwörter
Analytic function, starlike functions, uniformly convex functions

© 2015 Nihat Yağmur, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

On neighborhoods of functions associated with conic domains

Online veröffentlicht: 04. Apr. 2017

Seitenbereich: 291 - 302

Eingereicht: 08. Mai 2014

Akzeptiert: 30. Juni 2014

DOI: https://doi.org/10.1515/auom-2015-0020

SchlüsselwörterAnalytic function, starlike functions, uniformly convex functions

© 2015 Nihat Yağmur, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Schlüsselwörter
Analytic function, starlike functions, uniformly convex functions