We extended a theorem of Mishra and Srivastava (1983&#8211;1984) on 
<mml:math alttext="${|C,1|}_k$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo><mml:mi>C</mml:mi><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:msub><mml:mo>|</mml:mo><mml:mi>k</mml:mi></mml:msub></mml:math> summability factors, using almost increasing sequences,
to <mml:math alttext="$|\bar{N},p_n|_k$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo><mml:mover accent="true"><mml:mi>N</mml:mi><mml:mo>&#175;</mml:mo></mml:mover><mml:mo>,</mml:mo><mml:msub><mml:mi>p</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:msub><mml:mo>|</mml:mo><mml:mi>k</mml:mi></mml:msub></mml:math> summability under weaker conditions.

H&#252;seyin  Bor

An application of almost increasing sequences

We consider the class of univalent functions
 <mml:math alttext="$f(z) = z + a_{3}z^{3} + a_{4}z^{4} + \cdots$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>f</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mi>z</mml:mi><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>3</mml:mn></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:msub><mml:mi>a</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:msup><mml:mi>z</mml:mi><mml:mn>4</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mo>&#x22EF;</mml:mo></mml:math> analytic in the unit disc and
satisfying <mml:math alttext="$|(z^{2}f'(z)/f^{2}(z))-1|\deg1$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>f</mml:mi><mml:mo>&#x2032;</mml:mo></mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>|</mml:mo><mml:mo>&#x3C;</mml:mo><mml:mn>1</mml:mn></mml:math>, and show that such functions are starlike if they satisfy
<mml:math alttext="$|(z^{2}f'(z)/f^{2}(z)) - 1|\deg (1/\sqrt{2})$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>z</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msup><mml:mi>f</mml:mi><mml:mo>&#x2032;</mml:mo></mml:msup><mml:mrow><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msup><mml:mi>f</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn><mml:mo>|</mml:mo><mml:mo>&#x3C;</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:mrow><mml:msqrt><mml:mn>2</mml:mn></mml:msqrt></mml:mrow></mml:mrow></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:math>.

Vikramaditya Singh

On a class of univalent functions

Clifford and Preston (1961) showed several important characterizations of right groups. It was shown in Roy and So
(1998) that, among topological semigroups, compact right simple or left cancellative semigroups are in fact right groups, and the
closure of a right simple subsemigroup of a compact semigroup is always a right subgroup. In this paper, it is shown that such
results can be generalized in convergence semigroups. In the discussion of maximal right simple subsemigroups and maximal right
subgroups of semigroups, generalization of the results that no two maximal right simple subsemigroups and maximal right subgroups of a convergence semigroup intersect, is also established.

Phoebe Ho

Shing S. So

Right simple subsemigroups and right subgroups of compact convergence semigroups

We study the oscillatory behavior, the periodicity and the asymptotic behavior of the positive solutions of the system of two nonlinear difference equations <mml:math alttext="$x_{n + 1} = A + x_{n - 1}/y_n$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math> and  <mml:math alttext="$y_{n + 1} = A + y_{n-1}/x_n$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mrow><mml:mrow><mml:msub><mml:mi>y</mml:mi><mml:mrow><mml:mi>n</mml:mi><mml:mo>&#x2212;</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:mrow><mml:mo>/</mml:mo><mml:mrow><mml:msub><mml:mi>x</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:mrow></mml:mrow></mml:math>, where <mml:math alttext="$A$" id="E4" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> is a positive constant, and <mml:math alttext="$n = 0,1,\ldots$" id="E5" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>&#x2026;</mml:mo></mml:mrow></mml:math>.

G. Papaschinopoulos

C. J. Schinas

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Given a finite measure <mml:math alttext="$\mu$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#956;</mml:mi></mml:math>, we show that the set of norm
attaining bilinear forms is dense in the space of all continuous
bilinear forms on <mml:math alttext="$L_1(\mu)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>L</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>&#956;</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> if and only if <mml:math alttext="$\mu$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>&#956;</mml:mi></mml:math> is purely
atomic.

Yousef  Saleh

Norm attaining bilinear 
            forms on <svg style="vertical-align:-3.56265pt" height="16.625" version="1.1" viewBox="0 0 37.362499 16.625" width="37.362499" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D43F" d="M559 163q-23 -66 -68 -163h-474l6 26q62 4 79.5 19.5t28.5 75.5l78 409q7 35 8.5 49t-8 25t-24 13t-51.5 5l5 28h266l-6 -28q-65 -5 -79.5 -18t-25.5 -74l-76 -406q-10 -57 14 -75q12 -13 96 -13q93 0 126 29q41 40 76 109z" /></g><g transform="matrix(.012,-0,0,-.012,9.763,16.213)"><path id="x31" d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" /></g><g transform="matrix(.017,-0,0,-.017,16.1,12.138)"><path id="x28" d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" /></g><g transform="matrix(.017,-0,0,-.017,21.982,12.138)"><path id="x1D707" d="M538 96q-38 -45 -84 -76.5t-71 -31.5q-44 0 -15 117q12 49 18 89h-2q-99 -128 -163 -178q-36 -28 -70 -28q-26 0 -44 38h-2q-16 -69 -16 -129q0 -43 11 -75.5t24 -42.5l1 -6q-7 -12 -24.5 -23t-37.5 -11q-40 0 -40 76q0 52 35 202l94 407l78 24l5 -5q-32 -122 -72 -310&#xA;q-7 -33 0 -55t25 -22q36 0 105.5 75t107.5 145l32 146l75 26h10q-50 -197 -78 -347q-7 -38 6 -38q7 0 31 17t47 39z" /></g><g transform="matrix(.017,-0,0,-.017,31.416,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g></svg> 

Let <mml:math alttext="$A(n,m)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> be the number of solutions of <mml:math alttext="$\sum_{k=-n}^n \epsilon_k k=m$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mo>&#8721;</mml:mo><mml:mrow><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo>&#8722;</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mrow><mml:msub><mml:mi>&#1013;</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mi>k</mml:mi></mml:mrow><mml:mo>=</mml:mo><mml:mi>m</mml:mi></mml:math> where each <mml:math alttext="$\epsilon_k \in \{0,1\}$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>&#1013;</mml:mi><mml:mi>k</mml:mi></mml:msub><mml:mo>&#8712;</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:mrow><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mo>}</mml:mo></mml:mrow></mml:mrow></mml:math>. We determine the asymptotic behavior of <mml:math alttext="$A(n,m)$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>A</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:mi>n</mml:mi><mml:mo>,</mml:mo><mml:mi>m</mml:mi></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> for <mml:math alttext="$m = o(n^{3/2})$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mi>o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mrow><mml:msup><mml:mi>n</mml:mi><mml:mrow><mml:mrow><mml:mn>3</mml:mn><mml:mo>/</mml:mo><mml:mn>2</mml:mn></mml:mrow></mml:mrow></mml:msup></mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math>, extending results of van Lint and of Entringer.

Lane  Clark

On the representation of <svg style="vertical-align:-0.1638pt" height="7.9499998" version="1.1" viewBox="0 0 13.5375 7.9499998" width="13.5375" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D45A" d="M766 88q-40 -45 -83.5 -72.5t-63.5 -27.5q-39 0 -16 103l49 224q15 68 -10 68q-36 0 -112.5 -78.5t-105.5 -133.5q-27 -101 -38 -165q-35 -4 -77 -18l-6 6q42 153 70 288q12 55 10 78t-19 23q-38 0 -114 -80.5t-101 -131.5q-24 -68 -41 -165q-36 -3 -76 -18l-8 6&#xA;q56 202 87 347q7 33 -3 33q-8 0 -31.5 -17.5t-41.5 -35.5l-12 28q42 45 84 72t63 27q44 0 10 -124l-20 -75h2q92 121 193 178q36 21 64 21q62 0 28 -159l-8 -37h2q50 67 103.5 112t94.5 65q39 19 62 19q60 0 23 -156l-45 -189q-9 -39 2 -39q17 0 71 49z" /></g></svg> as <svg style="vertical-align:-3.71138pt" height="18.35" version="1.1" viewBox="0 0 61.787498 18.35" width="61.787498" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,13.675)"><path id="x2211" d="M697 -46l-42 -182h-602v25l325 457l-308 460v25h585l12 -154l-28 -5q-17 63 -39 88q-12 15 -36.5 20.5t-80.5 5.5h-277l268 -396l-311 -440h382q51 0 75.5 19.5t47.5 83.5z" /></g><g transform="matrix(.012,-0,0,-.012,12.812,5.388)"><path id="x1D45B" d="M495 86q-46 -47 -87 -72.5t-63 -25.5q-43 0 -16 107l49 210q7 34 8 50.5t-3 21t-13 4.5q-35 0 -109.5 -72.5t-115.5 -140.5q-21 -75 -38 -159q-50 -10 -76 -21l-6 8l84 340q8 35 -4 35q-17 0 -67 -46l-15 26q44 44 85.5 70.5t64.5 26.5q35 0 10 -103l-24 -98h2&#xA;q42 56 97 103.5t96 71.5q46 26 74 26q9 0 16 -2.5t14 -11.5t9.5 -24.5t-1 -44t-13.5 -68.5q-30 -117 -47 -200q-4 -19 -3.5 -25t6.5 -6q21 0 70 48z" /></g><g transform="matrix(.012,-0,0,-.012,12.812,18.138)"><path id="x1D458" d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1&#xA;l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" /></g><g transform="matrix(.012,-0,0,-.012,18.798,18.138)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.012,-0,0,-.012,25.782,18.138)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.012,-0,0,-.012,32.767,18.138)"><use xlink:href="#x1D45B"/></g><g transform="matrix(.017,-0,0,-.017,39.363,13.663)"><path id="x1D716" d="M401 397q0 -24 -24 -40q-18 -13 -31 3q-36 45 -93 45q-48 0 -86.5 -43t-51.5 -112l206 6l-8 -48l-203 6v-25q0 -68 26.5 -107t73.5 -39q73 0 148 64l16 -22q-40 -44 -91 -70.5t-102 -26.5q-75 0 -116.5 45t-41.5 136q0 125 99 208q36 30 87 50.5t93 20.5q46 0 72.5 -16&#xA;t26.5 -35z" /></g><g transform="matrix(.012,-0,0,-.012,46.575,17.737)"><use xlink:href="#x1D458"/></g><g transform="matrix(.017,-0,0,-.017,53.175,13.663)"><use xlink:href="#x1D458"/></g></svg> 

The periodic boundary value problems of a class of nonlinear differential equations are investigated.

Xiaojing  Yang

Periodic solutions of nonlinear differential equations

Commutative <mml:math alttext="$H^*$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>H</mml:mi><mml:mo>*</mml:mo></mml:msup></mml:math>-algebra is characterized in terms of the
property that the orthogonal complement of a right ideal is a left
ideal.

Parfeny P. Saworotnow

An exotic characterization of a commutative 
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In 1911, Steinhaus presented the following theorem: if <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> is a regular matrix then there exists a sequence of 0's and 1's which is
not <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math>-summable. In 1943, R. C. Buck characterized convergent
sequences as follows: a sequence <mml:math alttext="$x$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math> is convergent if and only if
there exists a regular matrix <mml:math alttext="$A$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>A</mml:mi></mml:math> which sums every subsequence of
<mml:math alttext="$x$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>x</mml:mi></mml:math>. In this paper, definitions for &#147;subsequences of a double
sequence&#148; and &#147;Pringsheim limit points&#148; of a double sequence are
introduced. In addition, multidimensional analogues of Steinhaus'
and Buck's theorems are proved.

Richard F. Patterson

Analogues of some fundamental theorems of summability theory

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International Journal of Mathematics and Mathematical Sciences publishes research across all fields of mathematics and mathematical sciences, such as pure and applied mathematics, mathematical physics, probability and mathematical statistics.

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International Journal of Mathematics and Mathematical Sciences

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International Journal of Mathematics and Mathematical Sciences