Fourth Hankel and Toeplitz determinant estimates for certain analytic functions associated with Four Leaf function

In this paper, we are introducing certain subfamilies of holomorphic functions and making an attempt to obtain an upper bound (UB) to the second and third order Hankel determinants by applying certain lemmas, Toeplitz determinants, for the normalized analytic functions belong to these classes, defined on the open unit disc in the complex plane. For one of the inequality, we have obtained sharp bound.

In this paper, we make an attempt to introduce a new subclass of analytic functions. Using the Toeplitz determinants, we obtain the best possible upper bound for the third-order Hankel determinant associated with the kth root transform [f(zk)]1/k of the normalized analytic function f(z) when it belongs to this class, defined on the open unit disc in the complex plane.

The objective of this paper is to obtain the best possible sharp upper bound for the second Hankel functional associated with the kth root transform [f(zk)]1/k of normalized analytic function f(z) when it belongs to certain subclass of analytic functions, defined on the open unit disc in the complex plane using Toeplitz determinants.

In this paper, we introduce and examine certain subclass $\ M_{q,\Sigma }\left( \varphi ,\beta \right) $ of analytic and bi-univalent functions on the open unit disk in the complex plane. Here, we give coefficient bound estimates, upper bound estimate for the second Hankel determinant and Fekete-Szegö inequality for the function belonging to this class. Some interesting special cases of the results obtained here are also discussed.

In this paper, we obtain the best possible upper bound to the second and third Hankel determinants for the functions belonging to a certain general subclass of analytic functions defined on the open unit disc in the complex plane using Toeplitz determinants. Relevant connections of the results presented here with those given in earlier works are also indicated.

The objective of this paper is to obtain an upper bound for the second Hankel functional associated with the k th root transform ... normalized analytic function f(z) belonging to starlike and convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.

The objective of this paper is to obtain sharp upper bound for the second Hankel functional associated with the k th root transform ( f(z k ) ) 1 k of normalized analytic function f(z) when it belongs to the class of starlike and convex functions with respect to symmetric points, dened on the open unit disc in the complex plane, using Toeplitz determinants.

The objective of this paper is to obtain sharp upper bound to the second Hankel functional associated with the k th root transform f(z k ) 1 k of normalized analytic function f(z) belonging to parabolic starlike and uniformly convex functions, defined on the open unit disc in the complex plane, using Toeplitz determinants.

In this paper, we investigate the coefficient bound estimates, second Hankel determinant, and Fekete-Szegö inequality for the analytic bi-univalent function class, which we call Mocanu type bi-starlike functions, related to a shell-shaped region in the open unit disk in the complex plane. Some interesting special cases of the results are also discussed.

The objective of this paper is to obtain an upper bound (not sharp) to the third order Hankel determinant for certain subclass of multivalent (p-valent) analytic functions, defined in the open unit disc E. Using the Toeplitz determinants, we may estimate the Hankel determinant of third kind for the normalized multivalent analytic functions belongng to this subclass. But, using the technique adopted by Zaprawa 1, i. e., grouping the suitable terms in order to apply Lemmas due to Hayami 2, Livingston 3 and Pommerenke 4, we observe that, the bound estimated by the method adopted by Zaprawa is more refined than using upon applying the Toeplitz determinants.

Starlikeness associated with certain strongly functions

The objective of this paper is to obtain an upper bound to Hankel determinant of third order for any function f, when it belongs to certain subclass of analytic functions, defined on the open unit disc in the complex plane.

Introduction. This introduction will present a quick survey of our results; the complete definitions necessary to state these results precisely are given in later sections. Let H°° denote the algebra of bounded analytic functions on the open unit disk in the complex plane. The maximal ideal space of H°° is denoted by M. We can think of the open unit disk as a dense subset of M. Carl Sundberg [11] proved that every function in BMO extends to a continuous function from M into the Riemann sphere; he also described several properties of these extensions. Sundberg was working in the context of functions of several real variables. In the next section of this paper we take advantage of the tools offered by analytic function theory to give considerably simpler proofs (in the context of one complex variable) of Sundberg's results about extensions of BMO functions. In the section of this paper on nontangential limits, we prove that a function on the disk that has a continuous extension to a small subset of M must have a nontangential limit at almost every point of the unit circle. We then use this result to produce a class of functions in the little Bloch space that cannot be extended to. be continuous functions from this small subset of M to the Riemann sphere. The section of this paper dealing with cluster sets and essential ranges shows how those sets can be computed from the appropriate continuous extensions. We use these results to give a new proof of Joel Shapiro's theorem [10] that for every function in VMO, the essential range equals the cluster set. The final section of the paper discusses some open questions.

Let D be the open unit disk in the complex plane C. The Bergman space L a 2 ( D ) is the Hilbert space of analytic functions f in D such that ‖ f ‖ 2 = ∫ D | f ( z ) | 2 d A ( z ) < ∞ where dA is the normalized area measure on D. If f ( z ) = ∑ n - 0 ∞ a n z n and g ( z ) = ∑ n - 0 ∞ b n z n are two functions in L a 2 ( D ) , then the inner product of f and g is given by 〈 f , g 〉 = ∫ D f ( z ) g ( z ) ¯ d A ( z ) = ∑ n = 0 ∞ a n b ¯ n n + 1 We study multiplication operators on L a 2 ( D ) induced by analytic functions. Thus for φ ∈ H ∞(D), the space of bounded analytic functions in D, we define M φ : L a 2 ( D ) → L a 2 ( D ) by M φ f = φ f , f ∈ L a 2 ( D ) It is easy to check that Mϕ is a bounded linear operator on L a 2 ( D ) with ‖Mφ‖=‖φ‖∞=sup{|φ(z)|:z∈D}.

The present extensive study is focused to find estimates for the upper bounds of the Toeplitz determinants. The logarithmic coefficients of univalent functions play an important role in different estimates in the theory of univalent functions, and in the this paper we derive the estimates of Toeplitz determinants and Toeplitz determinants of the logarithmic coefficients for the subclasses Ls𝑆𝑝𝑞 L𝑠C𝑝𝑞 and LsS𝑝𝑞 ∩ S, L𝑠𝐶𝑝𝑞 ∩ S, 0 q≤p≤10 𝑞≤𝑝≤1, defined by post quantum operators, which map the open unit disc 𝐷 onto the domain bounded by the limaçon curve defined by ∂Ds:={u+iv∈C:(u−1)2+v2−s4.2=4s2(u−1+s2)2+v2.}, where s∈−1,1.∖{0}. Keywords: Limaçon domain, subordination, (p, q)–derivative, Toeplitz and Hankel determinants, symmetric Toeplitz determinant, logarithmic coefficients, starlike functions with respect to symmetric points.