Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function

Frasin, B. A.; Swamy, S. R.; Nirmala, J.

doi:10.1007/s13370-020-00850-w

Cited by 26 publications

(14 citation statements)

References 14 publications

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“…Remark 13. For τ = 1, Corollary 12 reduces to a result of Frasin et al ( [30], Corollary 3.6). Further, allowing k = 1, we get Corollary 10 of Altnkaya [22].…”

Section: Corollary 10 If the Function G∈lmentioning

confidence: 80%

“…Remark 3. We note that (i) (i) When τ = 1, the family K 1 Σ ðμ, pk Þ was introduced by Frasin et al [30] (ii) The family L 1 Σ ð0, pk Þ ≡ S * Σ ðp k ÞÞ was mentioned by Güney et al [18], when μ = 0 and τ = 1 (iii) For μ = 0 and k = 1, the class L τ Σ ð0, p1 Þ ≡ S Σ ðp 1 Þ was investigated by Magesh et al [31] We now state the following lemma, which we will be using in the proof of our theorem.…”

Section: Introduction and Notationsmentioning

confidence: 99%

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A Comprehensive Family of Biunivalent Functions Defined by $k$ -Fibonacci Numbers

Frasin

Swamy²,

Aldawish

2021

Journal of Function Spaces

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By using k -Fibonacci numbers, we present a comprehensive family of regular and biunivalent functions of the type g z = z + ∑ j = 2 ∞ d j z j in the open unit disc D . We estimate the upper bounds on initial coefficients and also the functional of Fekete-Szegö for functions in this family. We also discuss few interesting observations and provide relevant connections of the result investigated.

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“…Remark 13. For τ = 1, Corollary 12 reduces to a result of Frasin et al ( [30], Corollary 3.6). Further, allowing k = 1, we get Corollary 10 of Altnkaya [22].…”

Section: Corollary 10 If the Function G∈lmentioning

confidence: 80%

Section: Introduction and Notationsmentioning

confidence: 99%

A Comprehensive Family of Biunivalent Functions Defined by $k$ -Fibonacci Numbers

Frasin

Swamy²,

Aldawish

2021

Journal of Function Spaces

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“…Generally, interest was shown to obtain the initial coefficient bounds and the celebrated inequality of Fekete-Szegö for the special subfamilies of Σ. Recently, the Horadam polynomial was used by Abirami et al [13] to find coefficient estimates for the families of bi-Bazilevic and λ-bistarlike function, Frasin et al [14] obtained coefficient estimates and Fekete-Szegö inequalities for certain subfamilies of Al-Oboudi-type bi-univalent functions related to k-Fibonacci numbers involving modified activation function, initial coefficient bounds for certain subsets of biunivalent functions family subordinate to Horadam polynomials were obtained in [15,16], Shaba and Wanas [17] obtained coefficient bounds which are sharp, for a family of bi-univalent functions using (U, V)-Lucas polynomials, Srivastava et al [18] have proposed a methodology to estimate coefficient bounds and Fekete-Szegö problem for certain subsets of bi-univalent function family linked with Horadam polynomials, and Swamy [19] and Swamy et al [20,21] have initiated the study of some subfamilies of bi-univalent function family subordinate to Horadam polynomials involving modified activation function. Swamy and Sailaja [22] have used Horadam polynomials to investigate coefficient estimates for two families of bi-univalent functions, Swamy et al [23] have introduced some subfamilies of Sȃlȃgean type biunivalent functions subordinate to (m, n)-Lucas polynomials and found initial coefficients, and Wanas and Alina [24] have fixed the Fekete-Szegö problem for Bazilevic biunivalent function class linked with Horadam polynomials.…”

Section: Preliminariesmentioning

confidence: 99%

Fekete–Szegö Inequality for Bi-Univalent Functions Subordinate to Horadam Polynomials

Shammaky

Frasin

Swamy³

2022

Journal of Function Spaces

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Making use of Horadam polynomials, we propose a special family of regular functions of the type g z = z + ∑ j = 2 ∞ d j z j which are bi-univalent (or bi-schlicht) in the disc z ∈ ℂ : z < 1 . We find estimates on the coefficients d 2 and d 3 and the functional of Fekete–Szegö for functions in this subfamily. Relevant connections to existing results and new observations of the main result are also presented.

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“…Linear activation functions are not widely applied in CNNs because of the discontinuous characteristics of their derivatives. In deep learning networks, nonlinear activation functions-Sigmoid, ReLU, Swish, Mish, and Logish-are frequently used [8,9]. Sigmoid maps all values to (0, 1), which is associated with the vanishing gradient problem.…”

Section: Introductionmentioning

confidence: 99%

Smish: A Novel Activation Function for Deep Learning Methods

2022

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Activation functions are crucial in deep learning networks, given that the nonlinear ability of activation functions endows deep neural networks with real artificial intelligence. Nonlinear nonmonotonic activation functions, such as rectified linear units, Tan hyperbolic (tanh), Sigmoid, Swish, Mish, and Logish, perform well in deep learning models; however, only a few of them are widely used in mostly all applications due to their existing inconsistencies. Inspired by the MB-C-BSIF method, this study proposes Smish, a novel nonlinear activation function, expressed as f(x)=x·tanh[ln(1+sigmoid(x))], which could overcome other activation functions with good properties. Logarithmic operations are first used to reduce the range of sigmoid(x). The value is then calculated using the tanh operator. Inputs are ultimately used to multiply the previous value, thus exhibiting negative output regularization. Experiments show that Smish tends to operate more efficiently than Logish, Mish, and other activation functions on EfficientNet models with open datasets. Moreover, we evaluated the performance of Smish in various deep learning models and the parameters of its function f(x)=αx·tanh[ln(1+sigmoid(βx))], and where α = 1 and β = 1, Smish was found to exhibit the highest accuracy. The experimental results show that with Smish, the EfficientNetB3 network exhibits a Top-1 accuracy of 84.1% on the CIFAR-10 dataset; the EfficientNetB5 network has a Top-1 accuracy of 99.89% on the MNIST dataset; and the EfficientnetB7 network has a Top-1 accuracy of 91.14% on the SVHN dataset. These values are superior to those obtained using other state-of-the-art activation functions, which shows that Smish is more suitable for complex deep learning models.

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Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function

Cited by 26 publications

References 14 publications

A Comprehensive Family of Biunivalent Functions Defined by $k$ -Fibonacci Numbers

A Comprehensive Family of Biunivalent Functions Defined by $k$ -Fibonacci Numbers

Fekete–Szegö Inequality for Bi-Univalent Functions Subordinate to Horadam Polynomials

Smish: A Novel Activation Function for Deep Learning Methods

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Some special families of holomorphic and Al-Oboudi type bi-univalent functions related to k-Fibonacci numbers involving modified Sigmoid activation function

Cited by 26 publications

References 14 publications

A Comprehensive Family of Biunivalent Functions Defined byk-Fibonacci Numbers

A Comprehensive Family of Biunivalent Functions Defined byk-Fibonacci Numbers

Fekete–Szegö Inequality for Bi-Univalent Functions Subordinate to Horadam Polynomials

Smish: A Novel Activation Function for Deep Learning Methods

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A Comprehensive Family of Biunivalent Functions Defined by $k$ -Fibonacci Numbers

A Comprehensive Family of Biunivalent Functions Defined by $k$ -Fibonacci Numbers