| Summary: | Here, we study strong differential subordinations for the extended new operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><msubsup><mi>R</mi><mrow><mi>λ</mi><mo>,</mo><mi>l</mi></mrow><mi>m</mi></msubsup></mrow></semantics></math></inline-formula> defined by the Hadamard product of the extended multiplier transformation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>l</mi></mfenced></mrow></semantics></math></inline-formula> and the extended Ruscheweyh derivative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>R</mi><mi>m</mi></msup></semantics></math></inline-formula>, on the class of normalized analytic functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi mathvariant="script">A</mi><mrow><mi>n</mi><mi>ζ</mi></mrow><mo>∗</mo></msubsup><mrow><mo>=</mo><mo>{</mo><mi>f</mi><mo>∈</mo><mi mathvariant="script">H</mi></mrow><mrow><mo>(</mo><mi>U</mi><mo>×</mo><mover><mi>U</mi><mo>¯</mo></mover><mo>)</mo></mrow><mo>,</mo><mspace width="4pt"></mspace><mi>f</mi><mrow><mo>(</mo><mi>z</mi><mo>,</mo><mi>ζ</mi><mo>)</mo></mrow><mo>=</mo><mi>z</mi><mo>+</mo><msub><mi>a</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mfenced open="(" close=")"><mi>ζ</mi></mfenced><msup><mi>z</mi><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mo>⋯</mo><mo>,</mo><mspace width="4pt"></mspace><mi>z</mi><mo>∈</mo><mi>U</mi><mo>,</mo></mrow></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>∈</mo><mover><mi>U</mi><mo>¯</mo></mover><mrow><mo>}</mo></mrow></mrow></semantics></math></inline-formula>, by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><msubsup><mi>R</mi><mrow><mi>λ</mi><mo>,</mo><mi>l</mi></mrow><mi>m</mi></msubsup><mo>:</mo><msubsup><mi mathvariant="script">A</mi><mrow><mi>n</mi><mi>ζ</mi></mrow><mo>∗</mo></msubsup><mo>→</mo><msubsup><mi mathvariant="script">A</mi><mrow><mi>n</mi><mi>ζ</mi></mrow><mo>∗</mo></msubsup></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><msubsup><mi>R</mi><mrow><mi>λ</mi><mo>,</mo><mi>l</mi></mrow><mi>m</mi></msubsup><mi>f</mi><mfenced separators="" open="(" close=")"><mi>z</mi><mo>,</mo><mi>ζ</mi></mfenced><mo>=</mo><mfenced separators="" open="(" close=")"><mi>I</mi><mfenced separators="" open="(" close=")"><mi>m</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>l</mi></mfenced><mo>∗</mo><msup><mi>R</mi><mi>m</mi></msup></mfenced><mi>f</mi><mfenced separators="" open="(" close=")"><mi>z</mi><mo>,</mo><mi>ζ</mi></mfenced><mo>.</mo></mrow></semantics></math></inline-formula>
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