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• W3201710276 abstract "Abstract In the current article, we examine some properties of analytic functions associated with cosine and exponential functions. We calculate some conditions on α so that; if <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mfrac> <m:mrow> <m:mi>α</m:mi> <m:msup> <m:mi>z</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mn>2</m:mn> <m:mo>−</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msup> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mi>p</m:mi> </m:mfrac> </m:math> $1+frac{alpha z^{2-p}f'(z)}{p}$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>α</m:mi> <m:mfrac> <m:mrow> <m:msup> <m:mi>z</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mi>f</m:mi> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mfrac> </m:math> $1+alpha frac{z^{2}f'(z)}{pf(z)}$ , <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>α</m:mi> <m:mfrac> <m:mrow> <m:msup> <m:mi>z</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>p</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:msup> <m:mi>f</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mfrac> </m:math> $1+alpha frac{z^{p+2}f'(z)}{pf^{2}(z)}$ and <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>α</m:mi> <m:mfrac> <m:mrow> <m:msup> <m:mi>z</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mn>2</m:mn> <m:mi>p</m:mi> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:msup> <m:mi>f</m:mi> <m:mo>′</m:mo> </m:msup> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:msup> <m:mi>f</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mn>3</m:mn> </m:mrow> </m:msup> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> </m:mfrac> </m:math> $1+alpha frac{z^{2p+2}f'(z)}{pf^{3}(z)}$ are subordinated by Janowski functions, then <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mfrac> <m:mrow> <m:mi>f</m:mi> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:msup> <m:mi>z</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mfrac> <m:mo>≺</m:mo> <m:mi>cos</m:mi> <m:mo>⁡</m:mo> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:math> $frac{f(z)}{z^{p}}prec cos (z)$ . Further, we also discuss the same implications for <m:math xmlns:m=http://www.w3.org/1998/Math/MathML> <m:mfrac> <m:mrow> <m:mi>f</m:mi> <m:mo stretchy=false>(</m:mo> <m:mi>z</m:mi> <m:mo stretchy=false>)</m:mo> </m:mrow> <m:msup> <m:mi>z</m:mi> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>p</m:mi> </m:mrow> </m:msup> </m:mfrac> <m:mo>≺</m:mo> <m:msup> <m:mrow class=MJX-TeXAtom-ORD> <m:mtext>e</m:mtext> </m:mrow> <m:mrow class=MJX-TeXAtom-ORD> <m:mi>z</m:mi> </m:mrow> </m:msup> </m:math> $frac{f(z)}{z^{p}}prec textrm{e}^{z}$ ." @default.
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• W3201710276 date "2021-10-01" @default.
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• W3201710276 title "Sufficient conditions for <i>p</i>-valent functions" @default.
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• W3201710276 doi "https://doi.org/10.1515/ms-2021-0040" @default.
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