For $-1\leq B \lt A\leq 1$, let $\mathcal{C}(A,B)$ denote the class of normalized Janowski convex functions defined in the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z| \lt 1\}$ that satisfy the subordination relation $1+zf''(z)/f'(z)\prec (1+Az)/(1+Bz)$. In the present article, we determine the sharp estimate of the Schwarzian norm for functions in the class $\mathcal{C}(A,B)$. The Dieudonné’s lemma which gives the exact region of variability for derivatives at a point of bounded functions, plays the key role in this study, and we also use this lemma to construct the extremal functions for the sharpness by a new method.

Let $\mathcal {K}_u$ denote the class of all analytic functions f in the unit disk $\mathbb {D}:=\{z\in \mathbb {C}:|z|<1\}$, normalised by $f(0)=f'(0)-1=0$ and satisfying $|zf'(z)/g(z)-1|<1$ in $\mathbb {D}$ for some starlike function g. Allu, Sokól and Thomas [‘On a close-to-convex analogue of certain starlike functions’, Bull. Aust. Math. Soc. 108 (2020), 268–281] obtained a partial solution for the Fekete–Szegö problem and initial coefficient estimates for functions in $\mathcal {K}_u$, and posed a conjecture in this regard. We prove this conjecture regarding the sharp estimates of coefficients and solve the Fekete–Szegö problem completely for functions in the class $\mathcal {K}_u$.

We obtain bounds for certain functionals defined on a class of meromorphic functions in the unit disc of the complex plane with a nonzero simple pole. These bounds are sharp in a certain sense. We also discuss possible applications of this result. Finally, we generalise the result to meromorphic functions with more than one simple pole.

This paper mainly considers the problem of generalizing a certain class of analytic functions by means of a class of difference operators. We consider some relations between starlike or convex functions and functions belonging to such classes. Some other useful properties of these classes are also considered.

Let $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$) functions and WLUD$^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$. Then, we show under what conditions a WLUD$^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$, that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$, for any $n\in \mathbb {N}$. Then, we formulate conditions under which a WLUD$^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point.

In this paper we provide some bounds for the quantity $\Vert f(y)-f(x)\Vert$, where $f:D\rightarrow \mathbb{C}$ is an analytic function on the domain $D\subset \mathbb{C}$ and $x$, $y\in {\mathcal{B}}$, a Banach algebra, with the spectra $\unicode[STIX]{x1D70E}(x)$, $\unicode[STIX]{x1D70E}(y)\subset D$. Applications for the exponential and logarithmic functions on the Banach algebra ${\mathcal{B}}$ are also given.

Suppose that the function $f$ is analytic in the open unit disk $\unicode[STIX]{x1D6E5}$ in the complex plane. For each $\unicode[STIX]{x1D6FC}>0$ a function $f^{[\unicode[STIX]{x1D6FC}]}$ is defined as the Hadamard product of $f$ with a certain power function. The function $f^{[\unicode[STIX]{x1D6FC}]}$ compares with the fractional derivative of $f$ of order $\unicode[STIX]{x1D6FC}$. Suppose that $f^{[\unicode[STIX]{x1D6FC}]}$ has a limit at some point $z_{0}$ on the boundary of $\unicode[STIX]{x1D6E5}$. Then $w_{0}=\lim _{z\rightarrow z_{0}}f(z)$ exists. Suppose that $\unicode[STIX]{x1D6F7}$ is analytic in $f(\unicode[STIX]{x1D6E5})$ and at $w_{0}$. We show that if $g=\unicode[STIX]{x1D6F7}(f)$ then $g^{[\unicode[STIX]{x1D6FC}]}$ has a limit at $z_{0}$.

The classical Remez inequality [‘Sur une propriété des polynomes de Tchebycheff’, Comm. Inst. Sci. Kharkov13 (1936), 9–95] bounds the maximum of the absolute value of a real polynomial $P$ of degree $d$ on $[-1,1]$ through the maximum of its absolute value on any subset $Z\subset [-1,1]$ of positive Lebesgue measure. Extensions to several variables and to certain sets of Lebesgue measure zero, massive in a much weaker sense, are available (see, for example, Brudnyi and Ganzburg [‘On an extremal problem for polynomials of $n$ variables’, Math. USSR Izv.37 (1973), 344–355], Yomdin [‘Remez-type inequality for discrete sets’, Israel. J. Math.186 (2011), 45–60], Brudnyi [‘On covering numbers of sublevel sets of analytic functions’, J. Approx. Theory162 (2010), 72–93]). Still, given a subset $Z\subset [-1,1]^{n}\subset \mathbb{R}^{n}$, it is not easy to determine whether it is ${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming (here ${\mathcal{P}}_{d}(\mathbb{R}^{n})$ is the space of real polynomials of degree at most $d$ on $\mathbb{R}^{n}$), that is, satisfies a Remez-type inequality: $\sup _{[-1,1]^{n}}|P|\leq C\sup _{Z}|P|$ for all $P\in {\mathcal{P}}_{d}(\mathbb{R}^{n})$ with $C$ independent of $P$. (Although ${\mathcal{P}}_{d}(\mathbb{R}^{n})$-norming sets are precisely those not contained in any algebraic hypersurface of degree $d$ in $\mathbb{R}^{n}$, there are many apparently unrelated reasons for $Z\subset [-1,1]^{n}$ to have this property.) In the present paper we study norming sets and related Remez-type inequalities in a general setting of finite-dimensional linear spaces $V$ of continuous functions on $[-1,1]^{n}$, remaining in most of the examples in the classical framework. First, we discuss some sufficient conditions for $Z$ to be $V$-norming, partly known, partly new, restricting ourselves to the simplest nontrivial examples. Next, we extend the Turán–Nazarov inequality for exponential polynomials to several variables, and on this basis prove a new fewnomial Remez-type inequality. Finally, we study the family of optimal constants $N_{V}(Z)$ in the Remez-type inequalities for $V$, as the function of the set $Z$, showing that it is Lipschitz in the Hausdorff metric.

The following rather surprising result is noted.

(1) A function f(z) = ∑anzn such that an ↓ 0 (n → ∞) belongs to H1 if and only if ∑(an/(n + 1)) < ∞.

A more subtle analysis is needed to prove that assertion (2) remains true if H1 is replaced by the predual, 1(⊂ H1), of the Bloch space. Assertion (1) extends the Hardy–Littlewood theorem, which says the following.

(2) f belongs to Hp (1 < p < ∞) if and only if ∑(n + 1)p−2anp < ∞.

A new proof of (2) is given and applications of (1) and (2) to the Libera transform of functions with positive coefficients are presented. The fact that the Libera operator does not map H1 to H1 is improved by proving that it does not map 1 into H1.

An ℝ-linear conjugation problem modelling the process of power fields forming in a heterogeneous infinite planar structure with an elliptical inclusion is considered. Exact analytical solutions are derived in the class of piece-wise meromorphic functions with their principal parts fixed. Cases with internal singularities and with singularities of the given principal parts at the interface are investigated.

It is shown that if f is an analytic function of sufficiently small exponential type in the right half-plane, which takes integer values on a subset of the positive integers having positive lower density, then f is a polynomial.

For Gauss–Turán quadrature formulae with an even weight function on the interval [−1, 1] and functions analytic in regions of the complex plane which contain in their interiors a circle of radius greater than I, the error term is investigated. In some particular cases we prove that the error decreases monotonically to zero. Also, for certain more general cases, we illustrate how to check numerically if this property holds. Some ℓ2-error estimates are considered.

A Bernstein–Walsh type inequality for ${{C}^{\infty }}$ functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak– Siciak theorem: a ${{C}^{\infty }}$ function on ${{\mathbb{R}}^{n}}$ that is real analytic on every line is real analytic; (2) Zorn–Lelong theorem: if a double power series $F\left( x,y \right)$ converges on a set of lines of positive capacity then $F\left( x,y \right)$ is convergent; (3) Abhyankar–Moh–Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero.

We consider a regenerative queueing process that is (partially) generated by an embedded phase-type renewal process. We show that, under some specified conditions, a performance measure is an analytic function of the rate of the renewal process. We then develop several methods for deriving its Taylor polynomial in the renewal rate. These polynomials are asymptotically exact as the rate decreases, and, thus, are called light traffic approximations of the performance measure. We show via examples that these new methods are not only more efficient compared to existing ones, but also more versatile due to their general settings, such as to conduct perturbation analysis and study transient behavior.

Let ψ be a functional of the sample path of a stochastic system driven by a Poisson process with rate λ . It is shown in a very general setting that the expectation of ψ,Eλ [ψ], is an analytic function of λ under certain moment conditions. Instead of following the straightforward approach of proving that derivatives of arbitrary order exist and that the Taylor series converges to the correct value, a novel approach consisting in a change of measure argument in conjunction with absolute monotonicity is used. Functionals of non-homogeneous Poisson processes and Wiener processes are also considered and applications to light traffic derivatives are briefly discussed.