On the shape-preserving properties of λ-Bernstein operators

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Su L., Mutlu G., Çekim B.

Journal of Inequalities and Applications, cilt.2022, sa.1, 2022 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 2022 Sayı: 1
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1186/s13660-022-02890-1
  • Dergi Adı: Journal of Inequalities and Applications
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Aerospace Database, Communication Abstracts, MathSciNet, Metadex, zbMATH, Directory of Open Access Journals, Civil Engineering Abstracts
  • Anahtar Kelimeler: lambda-Bernstein operators, Shape-preserving properties, Computer-aided design, Computer graphics
  • Gazi Üniversitesi Adresli: Evet

Özet

We investigate the shape-preserving properties of λ-Bernstein operators Bn,λ(f; x) that were recently introduced Bernstein-type operators defined by a new Beziér basis with shape parameter λ∈ [− 1 , 1]. For this purpose, we express Bn,λ(f; x) as a sum of a classical Bernstein operator and a sum of first order divided differences of f. Using this new representation, we prove that Bn,λ(f; x) preserves monotonic functions for all λ∈ [− 1 , 1]. However, we show by a counter example that Bn,λ(f; x) does not preserve convex functions for some λ∈ [− 1 , 1]. We present a weaker result for the case λ∈ [0 , 1] for a special class of functions. Finally, we analyze the monotonicity of λ-Bernstein operators with n and show that Bn,λ(f; x) is not monotonic with n for some λ if 1 / 2 < λ≤ 1.