Hadamard products of Schlicht functions and the Pólya-Schoenberg conjecture

Ruscheweyh, Stephan; Sheil‐Small, T.

doi:10.1007/bf02566116

Cited by 325 publications

(69 citation statements)

References 9 publications

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“…Finally, we state a result by Ruscheweyh and Sheil-Small ( [9], Lemma 2.7) concerning convolution of analytic functions.…”

Section: Background Resultsmentioning

confidence: 90%

Convolutions of planar harmonic convex mappings

Dorff

2001

Complex Variables, Theory and Application: An International Jou

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Abstract. Ruscheweyh and Sheil-Small proved that convexity is preserved under the convolution of univalent analytic mappings in K. However, when we consider the convolution of univalent harmonic convex mappings in K O H , this property does not hold. In fact, such convolutions may not be univalent. We establish some results concerning the convolution of univalent harmonic convex mappings provided that it is locally univalent. In particular, we show that the convolution of a right half-plane mapping in K O H with either another right halfplane mapping or a vertical strip mapping in K O H is convex in the direction of the real axis. Further, we give a condition under which the convolution of a vertical strip mapping in K O H with itself will be convex in the direction of the real axis.

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“…Finally, we state a result by Ruscheweyh and Sheil-Small ( [9], Lemma 2.7) concerning convolution of analytic functions.…”

Section: Background Resultsmentioning

confidence: 90%

Convolutions of planar harmonic convex mappings

Dorff

2001

Complex Variables, Theory and Application: An International Jou

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“…To prove Theorem 4, we take advantage of the several convolution results due mainly to Ruscheweyh [11], [12]. Let f ∈ J * (σ, α β ; A, B), then it easily follows that…”

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confidence: 99%

“…is convex univalent ( [11]), and in the famous paper [12], it was proved as a result of the Pólya-Schoenberg conjecture that the functions which are convex univalent have the same property preserved under a convolution structure also. Accordingly, under our assumptions on α, β and p, the function (47)…”

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confidence: 99%

Certain results on subordinations associated with a differintegral operator

Dziok¹,

Raina²,

Sokół³

2011

MATH. SCAND.

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“…The third condition we want to cite seems to be the most famous, since it was used by Ruscheweyh and Sheil-Small in [22] to prove the Pólya-Schoenberg conjecture. This one is as follows.…”

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confidence: 99%