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The purpose of this paper is to find coefficient estimates for the class of functions ℳ N (γ , ϑ , λ) consisting of analytic functions f normalized by f(0) = f′(0) – 1 = 0 in the open unit disk D subordinated to a function generated using the van der Pol numbers, and to derive certain coefficient estimates for a2, a3, and the Fekete-Szegő functional upper bound for f ∈ ℳ N (γ , ϑ , λ) . Similar results were obtained for the logarithmic coefficients of these functions. Further application of our results to certain functions defined by convolution products with a normalized analytic functions is given, and in particular, we obtain Fekete-Szegő inequalities for certain subclasses of functions defined through the Poisson distribution series. [ABSTRACT FROM AUTHOR]

In this paper, we study the higher-order Schwarzian derivative Sn(f) proposed by H. Tamanoi [Higher Schwarzian operators and combinatorics of the Schwarzian derivative, Math. Ann. 305 (1996), 127–151]. For the strongly starlike functions of order α and strongly convex functions of order α, the sharp bound of |S3(f)(0)| is obtained. When n ∈ [2, 7], we prove that the higher Bers maps induced by Sn(f) on Weil-Petersson Teichmüller space and BMO-Teichmüller space are holomorphic. [ABSTRACT FROM AUTHOR]

In this paper, an equality is proved for twice differentiable convex functions involving Riemann–Liouville fractional integral. With the help of this equality, there are established several fractional Simpson type inequalities for functions whose second derivatives in absolute value are convex. By using special cases of the main results, previously obtained Simpson type inequalities are found for the Riemann–Liouville fractional integral. [ABSTRACT FROM AUTHOR]

In [Logarithmic coefficient bounds and coefficient conjectures for classes associated with convex functions, J. Funct. Spaces 2021 (2021), Art. ID 6690027], Alimohammadi et al. presented a few conjectures for the logarithmic coefficients γn of the functions f belonging to some well-known classes like C (1 + α z) for α ∈ (0, 1], and C V h p l (1 / 2). For example, it is conjectured that if the function f ∈ C (1 + α z) , then the logarithmic coefficients of f satisfy the inequalities | γ n | ≤ α 2 n (n + 1) , n ∈ N. Equality is attained for the function Lα, n, that is, log L α , n (z) z = 2 ∑ n = 1 ∞ γ n (L α , n) z n = α n (n + 1) z n + ... , z ∈ U. The aim of this paper is to confirm that these conjectures hold for the coefficient γn0−1 whenever the function f has the form f (z) = z + ∑ k = n 0 ∞ a k z k , z ∈ U for some n 0 ∈ N , n0⩾2. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some norm inequalities involving convex and concave functions, which are the generalizations of the classical Clarkson inequalities. Let A1, . . . , An be bounded linear operators on a complex separable Hilbert space H and let α1, . . ., αn be positive real numbers such that ∑j=1nαj = 1. We show that for every unitarily invariant norm, If ƒ is a non-negative function on [0, ∞) such that ƒ(0) = 0 and g(t) = ƒ(t) is convex, then |||∑j=1nαjƒ(|Aj|)||| ≥ |||∑j,k∈Sℓ (ƒ(√αjαk /4αℓ(1−αℓ)| Aj + Ak − 2 ∑j=1nαjAj|) + ƒ(√αjαk(2αℓ−1)/4αℓ(1−αℓ)| Aj − Ak|)) + ƒ(|∑j=1nαjAj|)||| for ℓ = 1, . . . , n. If ƒ is a non-negative function on [0, ∞) such that g(t) = ƒ(√t) is concave, then the inverse inequality holds. Here, the symbol Sℓ = {1, . . . , n} ∖ {ℓ} for ℓ ∈ {1, . . . , n}. In addition, we provide some applications of the above inequalities. [ABSTRACT FROM AUTHOR]

For normalized starlike functions f : 픻 → ℂ, we consider the analytic functions g : 픻 → ℂ defined by g(z) = (1 + z(f″(z))/f′(z))/(zf′(z)/f(z)) and g(z) = (1 − α)(zf′(z))/f(z) + α(1 + (zf″(z))/f′(z)), 0 ≤ α ≤ 1. We determine the largest radius ρ with 0 < ρ ≤ 1 such that g(ρ z) is subordinate to various functions with positive real part. [ABSTRACT FROM AUTHOR]

We generalize cyclic Jensen's inequality utilizing the theory of 4-convex function under the effect of introduced Green functions. We formulate result for power means and quasi means. Also we give applications in information theory by giving new estimations of generalized Csiszár divergence, Rényi-divergence, Shannon-entropy, Kullback-Leibler divergence and χ2-divergence. [ABSTRACT FROM AUTHOR]

We investigate the non-univalent function's properties reminiscent of the theory of univalent starlike functions. Let the analytic function ψ (z) = ∑ i = 1 ∞ A i z i , A1 ≠ 0 be univalent in the unitdisk. Non-univalent functions may be found in the class ℱ (ψ) of analytic functions f of the form f (z) = z + ∑ k = 2 ∞ a k z k satisfying (zf′ (z)/f (z) – 1) ≺ ψ(z). Such functions, like the Ma and Minda classes k=2 of starlike functions, also have nice geometric properties. For these functions, growth and distortion theorems have been established. Further, we obtain bounds for some sharp coefficient functionals and establish the Bohr and Rogosinki phenomenon for the class ℱ (ψ) . Non-analytic functions that share properties of analytic functions are known as poly-analytic functions. Moreover, we compute Bohr and Rogosinski's radius for poly-analytic functions with analytic counterparts in the class ℱ (ψ) or classes of Ma-Minda starlike and convex functions. [ABSTRACT FROM AUTHOR]