<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we define a new subclass of bi-univalent functions involving <italic>q</italic>-difference operator in the open unit disk. For functions belonging to this class, we obtain estimates on the first two Taylor-Maclaurin coefficients <italic>|a</italic>2<italic>|</italic> and <italic>|a</italic>3<italic>|</italic>.</abstract>

Bi-Bazilevič functions of complex order involving Ruscheweyh type <italic>q</italic>-difference operator

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we study <italic>∗</italic>-g-frames in tensor products of Hilbert <italic>C∗</italic>-modules. We show that a tensor product of two <italic>∗</italic>-g-frames is a <italic>∗</italic>-g-frame, and we get some result.</abstract>

*-g-frames in tensor products of Hilbert <italic>C*</italic>-modules

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, we define a new subclass of bi-univalent functions involving a Hohlov operator in the open unit disk. For functions belonging to this class, we obtain estimates on the first two Taylor-Maclaurin coefficients |<italic>a</italic>2<italic>|</italic> and <italic>|a</italic>3<italic>|</italic>.</abstract>

Department of Mathematics School of Advanced Sciences

Faculty of Mathematics and Computer Science,

Estimate for initial MacLaurin coefficients of certain subclasses of bi-univalent functions of complex order associated with the Hohlov operator

<abstract xmlns="http://www.w3.org/1999/xhtml">Each family <italic>ℳ</italic> of means has a natural, partial order (point-wise order), that is <italic>M ≤ N</italic> iff <italic>M</italic>(<italic>x</italic>) <italic>≤ N</italic>(<italic>x</italic>) for all admissible <italic>x</italic>.In this setting we can introduce the notion of interval-type set (a subset <italic>ℐ ⊂ℳ</italic> such that whenever <italic>M ≤ P ≤ N</italic> for some <italic>M, N ∈ℐ</italic> and <italic>P ∈ℳ</italic> then <italic>P ∈ℐ</italic>). For example, in the case of power means there exists a natural isomorphism between interval-type sets and intervals contained in real numbers. Nevertheless there appear a number of interesting objects for a families which cannot be linearly ordered.In the present paper we consider this property for Gini means and Hardy means. Moreover, some results concerning <italic>L</italic>∞ metric among (abstract) means will be obtained.</abstract>

Interval-type theorems concerning means

<abstract xmlns="http://www.w3.org/1999/xhtml">We study three properties associated to the recurrence of orbits in flows: asymptotic periodicity, positive asymptotic periodicity and G-asymptotic periodicity. We determine which implications between these notions hold and which do not. We also show how these notions are related to Lyapunov stability.</abstract>

On asymptotically periodic-like motions in flows

<abstract xmlns="http://www.w3.org/1999/xhtml">Let <italic>Zn</italic> = <italic>p</italic>0 + <italic>p</italic>1 + <italic>···</italic> + <italic>pn</italic> be a configuration of points in ℙ2, where all points <italic>pi</italic> except <italic>p</italic>0 lie on a line, and let <italic>I</italic>(<italic>Zn</italic>) be its corresponding homogeneous ideal in 𝕂 [ℙ2]. The resurgence and the Waldschmidt constant of <italic>I</italic>(<italic>Zn</italic>) in [5] have been computed. In this note, we compute these two invariants for the defining ideal of a fat point subscheme <italic>Zn,c</italic> = <italic>cp</italic>0 + <italic>p</italic>1 +<italic>···</italic> + <italic>pn</italic>, i.e. the point <italic>p</italic>0 is considered with multiplicity <italic>c</italic>. Our strategy is similar to [5].</abstract>

Resurgence and Waldschmidt constant of the ideal of a fat almost collinear subscheme in ℙ2

<abstract xmlns="http://www.w3.org/1999/xhtml">It has been a long-standing problem posed by the first author in a conference in Marseille in 1990 to characterize semitopological semigroups which have common fixed point property when acting on a nonempty weak* compact convex subset of a dual Banach space as weak* continuous and norm nonexpansive mappings. Our investigation in the paper centers around this problem. Our main results rely on the well-known Ky Fan’s inequality for convex functions.</abstract>

Department of Mathematical and Statistical Sciences

Fixed point properties for semigroups of nonexpansive mappings on convex sets in dual Banach spaces

<abstract xmlns="http://www.w3.org/1999/xhtml">In this article the pseudo-amenability and pseudo-contractibility of restricted semigroup algebra <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mi>l</mi></mrow><mi>r</mi><mn>1</mn></msubsup><mo>(</mo><mi>S</mi><mo>)</mo></mrow></math><tex-math>$l_r^1(S)$</tex-math></alternatives></inline-formula> and semigroup algebra, <italic>l</italic>1(<italic>Sr</italic>) on restricted semigroup, <italic>Sr</italic> are investigated for different classes of inverse semi-groups such as Brandt semigroup, and Clifford semigroup. We particularly show the equivalence between pseudo-amenability and character amenability of restricted semigroup algebra on a Clifford semigroup and semigroup algebra on a restricted semigroup. Moreover, we show that when <italic>S</italic> = <italic>M</italic>0(<italic>G, I</italic>)is a Brandt semigroup, pseudo-amenability of <italic>l</italic>1(<italic>Sr</italic>) is equivalent to its pseudo-contractibility.</abstract>

Pseudo-amenability and pseudo-contractibility of restricted semigroup algebra

<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper we study existence and uniqueness of solutions for a coupled system consisting of fractional differential equations of Caputo type, subject to Riemann–Liouville fractional integral boundary conditions. The uniqueness of solutions is established by Banach contraction principle, while the existence of solutions is derived by Leray–Schauder’s alternative. We also study the Hyers–Ulam stability of mentioned system. At the end, examples are also presented which illustrate our results.</abstract>

Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions

<abstract xmlns="http://www.w3.org/1999/xhtml">The convex hull of the subgraph of the prime counting function <italic>x → π</italic>(<italic>x</italic>) is a convex set, bounded from above by a graph of some piecewise affine function <italic>x →</italic> (<italic>x</italic>). The vertices of this function form an infinite sequence of points <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mrow><mrow><mo>(</mo><msub><mrow><mi>e</mi></mrow><mi>k</mi></msub><mo>,</mo><mi>π</mi><mo>(</mo><msub><mrow><mi>e</mi></mrow><mi>k</mi></msub><mo>)</mo><mo>)</mo></mrow></mrow><mn>1</mn><mo>∞</mo></msubsup></mrow></math><tex-math>$({e_k},\pi ({e_k}))_1^\infty $</tex-math></alternatives></inline-formula>. The elements of the sequence (ek)1∞ shall be called <italic>the extremal prime numbers</italic>. In this paper we present some observations about the sequence (ek)1∞ and we formulate a number of questions inspired by the numerical data. We prove also two – it seems – interesting results. First states that <italic>if the Riemann Hypothesis is true, then</italic><inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mrow><mfrac><mrow><msub><mrow><mi>e</mi></mrow><mi>k</mi></msub><mo>+</mo><mn>1</mn></mrow><mrow><msub><mrow><mi>e</mi></mrow><mi>k</mi></msub></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></math><tex-math>${{{e_k} + 1} \over {{e_k}}} = 1$</tex-math></alternatives></inline-formula>. The second, also depending on Riemann Hypothesis, describes the order of magnitude of the differences between consecutive extremal prime numbers.</abstract>

Prime numbers with a certain extremal type property

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Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

Volume 17 (2018): Issue 1 (December 2018)

Bi-Bazilevič functions of complex order involving Ruscheweyh type q-difference operator

-g-frames in tensor products of Hilbert C-modules