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Convolution theorems for starlike and convex functions in the unit disc

100%

Anbudurai M., Parvatham R., Ponnusamy S., Singh V.

Annales Polonici Mathematici

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2004

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tom 84

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nr 1

27-39

EN

Let A denote the space of all analytic functions in the unit disc Δ with the normalization f(0) = f'(0) − 1 = 0. For β < 1, let $P⁰_{β} = {f ∈ A: Re f'(z) > β, z ∈ Δ}$. For λ > 0, suppose that 𝓕 denotes any one of the following classes of functions: $M^{(1)}_{1,λ} = {f ∈ 𝓐 : Re{z(zf'(z))''} > -λ, z ∈ Δ}$, $M^{(2)}_{1,λ} = {f ∈ 𝓐 : Re{z(z²f''(z))''} > -λ, z ∈ Δ}$, $M^{(3)}_{1,λ} = {f ∈ 𝓐 : Re{1/2 (z(z²f'(z))'')' - 1} > -λ, z ∈ Δ}$. The main purpose of this paper is to find conditions on λ and γ so that each f ∈ 𝓕 is in $𝓢_{γ}$ or $𝒦_{γ}$, γ ∈ [0,1/2]. Here $𝓢_{γ}$ and $𝒦_{γ}$ respectively denote the class of all starlike functions of order γ and the class of all convex functions of order γ. As a consequence, we obtain a number of convolution theorems, namely the inclusions $M_{1,α} ∗ 𝓖 ⊂ 𝓢_{γ}$ and $M_{1,α} ∗ 𝓖 ⊂ 𝒦_{γ}$, where 𝓖 is either $𝓟⁰_{β}$ or $M_{1,β}$. Here $M_{1,λ}$ denotes the class of all functions f in 𝓐 such that Re(zf''(z)) > -λ for z ∈ Δ.

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On a Mocanu type generalization of the Kaplan class K(α, β) of analytic functions

100%

Padmanabhan K. S., Parvatham R., Radha S.

Annales Polonici Mathematici

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1988-1989

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tom 49

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nr 1

35-44

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On certain generalized close-to-star functions in the unit disc

99%

Padmanabhan K. S., Parvatham R.

Annales Polonici Mathematici

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1980

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tom 37

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nr 1

1-11

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Properties of a class of functions with bounded boundary rotation

99%

Padmanabhan K. S., Parvatham R.

Annales Polonici Mathematici

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1975-1976

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tom 31

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nr 3

311-323

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On certain classes of analytic functions

62%

Parvatham R., Radha S.

Annales Polonici Mathematici

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1988-1989

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tom 49

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nr 1

31-34

Ograniczanie wyników

5 Annales Polonici Mathematici

5 Parvatham R.

3 Padmanabhan K. S.

2 Radha S.

1 Anbudurai M.

1 Ponnusamy S.

1 Singh V.

1 2004

2 1988

1 1980

1 1976

Convolution theorems for starlike and convex functions in the unit disc

On a Mocanu type generalization of the Kaplan class K(α, β) of analytic functions

On certain generalized close-to-star functions in the unit disc

Properties of a class of functions with bounded boundary rotation

On certain classes of analytic functions