<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper we have introduced and studied the subclass ℛ𝒥 (<italic>d</italic>, <italic>α</italic>, <italic>β</italic>) of univalent functions defined by the linear operator <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2015-0001_eq_g001.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="italic">RI</mi><mrow><mi>n</mi><mo>,</mo><mo>λ</mo><mo>,</mo><mi>l</mi></mrow><mo>γ</mo></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math><tex-math>$RI_{n,\lambda ,l}^\gamma f(z)$</tex-math></alternatives></inline-formula> defined by using the Ruscheweyh derivative <italic>Rnf</italic>(<italic>z</italic>) and multiplier transformation <italic>I</italic> (<italic>n</italic>, λ, <italic>l</italic>) <italic>f</italic>(<italic>z</italic>), as <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2015-0001_eq_g002.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="italic">RI</mi><mrow><mi>n</mi><mo>,</mo><mo>λ</mo><mo>,</mo><mi>l</mi></mrow><mo>γ</mo></msubsup><mo>:</mo><mo>𝒜</mo><mo>→</mo><mo>𝒜</mo></math><tex-math>$RI_{n,\lambda ,l}^\gamma :{\cal A} \to {\cal A}$</tex-math></alternatives></inline-formula>, <inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_auom-2015-0001_eq_g003.png"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><msubsup><mi mathvariant="italic">RI</mi><mrow><mi>n</mi><mo>,</mo><mo>λ</mo><mo>,</mo><mi>l</mi></mrow><mo>γ</mo></msubsup><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mo>γ</mo><mo stretchy="false">)</mo><msup><mi>R</mi><mi>n</mi></msup><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>+</mo><mo>γ</mo><mi>I</mi><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mo>λ</mo><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo><mi>f</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></math><tex-math>$RI_{n,\lambda ,l}^\gamma f(z) = (1 - \gamma )R^n f(z) + \gamma I(n,\lambda ,l)f(z)$</tex-math></alternatives></inline-formula>, <italic>z</italic> ∈ <italic>U</italic>, where 𝒜<italic>n</italic> ={<italic>f</italic> ∈ ℋ(<italic>U</italic>) : <italic>f</italic>(<italic>z</italic>) = <italic>z</italic> + <italic>an</italic>+1<italic>zn</italic>+1 + . . . , <italic>z</italic> ∈ <italic>U</italic>}is the class of normalized analytic functions with 𝒜1 = 𝒜. The main object is to investigate several properties such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and the radii of starlikeness, convexity and close-to-convexity of functions belonging to the class ℛ𝒥(<italic>d</italic>, <italic>α</italic>, <italic>β</italic>).</abstract>

In this paper we have introduced and studied the subclass ℛ𝒥 (d, α, β) of univalent functions defined by the linear operator RIn,λ,lγf(z)$RI_{n,\lambda ,l}^\gamma f(z)$ defined by using the Ruscheweyh derivative Rnf(z) and multiplier transformation I (n, λ, l) f(z), as RIn,λ,lγ:𝒜→𝒜$RI_{n,\lambda ,l}^\gamma :{\cal A} \to {\cal A}$, RIn,λ,lγf(z)=(1−γ)Rnf(z)+γI(n,λ,l)f(z)$RI_{n,\lambda ,l}^\gamma f(z) = (1 - \gamma )R^n f(z) + \gamma I(n,\lambda ,l)f(z)$, z ∈ U, where 𝒜n ={f ∈ ℋ(U) : f(z) = z + an+1zn+1 + . . . , z ∈ U}is the class of normalized analytic functions with 𝒜1 = 𝒜. The main object is to investigate several properties such as coefficient estimates, distortion theorems, closure theorems, neighborhoods and the radii of starlikeness, convexity and close-to-convexity of functions belonging to the class ℛ𝒥(d, α, β).

Properties on a subclass of univalent functions defined by using a multiplier transformation and Ruscheweyh derivative

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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

In this paper we have introduced and studied the subclass ℛ𝒥 (d, α, β) of univalent functions defined by the linear operator RIn,λ,lγf(z)$RI_{n,\lambda...

Properties on a subclass of univalent functions defined by using a multiplier transformation and Ruscheweyh derivative

Published Online: Apr 04, 2017

Page range: 9 - 24

Received: May 03, 2014

Accepted: Jun 29, 2014

DOI: https://doi.org/10.1515/auom-2015-0001

Keywords
univalent function, Starlike functions, Convex functions, Distortion theorem

© 2015 Alina Alb Lupaş, published by De Gruyter Open

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

Properties on a subclass of univalent functions defined by using a multiplier transformation and Ruscheweyh derivative

Published Online: Apr 04, 2017

Page range: 9 - 24

Received: May 03, 2014

Accepted: Jun 29, 2014

DOI: https://doi.org/10.1515/auom-2015-0001

Keywordsunivalent function, Starlike functions, Convex functions, Distortion theorem

© 2015 Alina Alb Lupaş, published by De Gruyter Open

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

Keywords
univalent function, Starlike functions, Convex functions, Distortion theorem